TOPICS IN COMPLEX ANALYSIS \ centerlineBANACH CENTER{TOPICS PUBLICATIONS IN COMPLEX ANALYSIS comma VOLUME} 3 1 INSTITUTE OF MATHEMATICS \noindentPOLISH ACADEMYBANACH CENTER OF SCIENCES PUBLICATIONS , VOLUME 3 1 WARSZAWA 1 9 9 5 TOPICS IN COMPLEX ANALYSIS \ centerlineTHE USE OF{INSTITUTE D hyphen MODULES OF MATHEMATICS TO .. STUDY} BANACH CENTER PUBLICATIONS , VOLUME 3 1 EXPONENTIAL POLYNOMIALS INSTITUTE OF MATHEMATICS \ centerlineC period ..{ APOLISH period .. ACADEMY B ERENSTEIN OF SCIENCES } POLISH ACADEMY OF SCIENCES Mathematics Depa to the power of r-t ment ampersand Institute of Systems Research comma .. University of Maryland WARSZAWA 1 9 9 5 \ centerlineCollege Park{WARSZAWA comma Maryland 1 9 9 20742 5 } comma .. U period S period A period THE USE OF D− MODULES TO STUDY A period .. YGER EXPONENTIAL POLYNOMIALS \ centerlineD acute-e pa{THE to the USE power OF of t-r$ D ement− de$ Math MODULES acute-e TO matiques\quad commaSTUDY ..} Universit e-acute de Bordeaux I C . A . B ERENSTEIN 334 5 Talence comma France Mathematics Depar−t ment & Institute of Systems Research , University of Maryland \ centerlineAbstract period{EXPONENTIAL .. This is a summary POLYNOMIALS of recent} work where we introduced a class of D hyphen modules College Park , Maryland 20742 , U . S . A . adapted to study ideals generated by exponential polynomials period A . YGER \ centerline0 period .. Introduction{C. \quad periodA. ..\quad This isB an ERENSTEIN expanded version} of the lecture given by the first D epa´ t−r ement de Math e´ matiques , Universit e´ de Bordeaux I author at the Banach Center comma during the workshop on residues comma November 1 992 period 334 5 Talence , France \ centerlineWe introduce{Mathematics a new method t o $ study Depa ideals ˆ{ r generated−t }$ mentby exponential $ \& $ polynomials Institute comma of Systems Research , \quad University of Maryland } Abstract . This is a summary of recent work where we introduced a class of D− modules inspired by the theory on D hyphen modules open square bracket 1 6 comma 1 7 comma 1 8 closing square bracket adapted to study ideals generated by exponential polynomials . period\ centerline .... Detailed{ College proofs of Park some ,of Maryland the 20742 , \quad U.S.A. } 0 . Introduction . This is an expanded version of the lecture given by the first statements of this paper can b e found in open square bracket 14 closing square bracket period .... We take the author at the Banach Center , during the workshop on residues , November 1 992 . opportunity\ centerline t o{ thankA. \quad YGER } We introduce a new method t o study ideals generated by exponential polynomials , Professors Jak o-acute b czak comma Ple acute-s niak comma and Aizenberg for their hospitality period inspired by the theory on D− modules [ 1 6 , 1 7 , 1 8 ] . Detailed proofs of some of the \ centerlineLet us recall{D that $ an\ exponentialacute{e} polynomialpa ˆ{ t f− ofr n} complex$ ement variables de Math with $ \acute{e} $ matiques , \quad U n i v e r s i t statements of this paper can b e found in [ 14 ] . We take the opportunity t o thank $ \frequenciesacute{e} in$ a finitely de Bordeaux generated I subgroup} Capital Gamma of C to the power of n i s a function of the form Professors Jako ´ b czak , Ples ´ niak , and Aizenberg for their hospitality . Line 1 f open parenthesis z sub 1 comma period period period comma z sub n closing parenthesis = f open parenthesis Let us recall that an exponential polynomial f of n complex variables with z\ centerline closing parenthesis{334 = 5 sum Talence p sub gamma , France open parenthesis} z closing parenthesis exponent open parenthesis gamma times frequencies in a finitely generated subgroup Γ of n i s a function of the form z closing parenthesis comma Line 2 gamma in Capital Gamma C \ hspace ∗{\ f i l l } Abstract . \quad This is a summaryX of recent work where we introduced a class of where the sum is finite comma the p subf(z gamma, ..., z ) are= f polynomials(z) = p ( commaz) exp(γ ....· z) and, gamma times z = gamma 1 to the power$ D of− z 1$ plus modules period period period plus gamma1 n n to the power ofγ z n period Such a function belongs t o the algebra A sub phi open parenthesis C to theγ ∈ powerΓ of n closing parenthesis .. of entire \noindent adapted to study ideals generated by exponential polynomials . functions F satisfying z z where the sum is finite , the pγ are polynomials , and γ · z = γ1 1 + ... + γn n. t h-line sub e growth condition : n \ hspace ∗{\ fSuch i l l }0 a function . \quad belongsIntroduction t o the algebra . \Aquadφ(C )This of entire is an functions expandedF satisfying version t h − ofline thee lecture given by the first exists C greatergrowth 0 bar condition F open : parenthesis z closing parenthesis bar less or equal C exponent open parenthesis C phi open parenthesis z closing parenthesis closing parenthesis comma \noindent author at the Banach Center , during the workshop on residues , November 1 992 . 1 99 1 Mathematics Subject Classification∃C : 32 > A0 1| 5F comma(z) |≤ C 32exp( A 25Cφ comma(z)), 34 K 5 comma 34 K 35 comma 32 S 40 period \noindentResearch ofWe the1 introduce 99first 1 Mathematics author apartially new Subject method supported Classification t by o the study: NSF 32 A ideals grants 1 5 , 32 DMS A generated 25hyphen , 34 K 5 9000619 , by34 K exponential 35 and , 32 CDR S 40 . hyphen polynomials , 880301 2 periodResearch of the first author partially supported by the NSF grants DMS - 9000619 and CDR - \noindentResearch ofinspired880301 the second 2 . by author the partially theory supported on $ D by the− NSF$ modules grant DMS [ hyphen 16 ,17 9000619 ,18 period ] . \ h f i l l Detailed proofs of some of the The paper is in finalResearch form ofand the no second version author of it partially will be published supported elsewhereby the NSF period grant DMS - 9000619 . \noindentopen squarestatements bracket 77The closing of paper this square is in paper final bracket form can and b no e version found of it in will [ be 14 published ] . \ h elsewhere f i l l We . take the opportunity t o thank

\noindent Professors Jak $ \acute{o} $ bczak[77] , Ple $ \acute{ s } $ niak , and Aizenberg for their hospitality .

\ hspace ∗{\ f i l l } Let us recall that an exponential polynomial $ f $ of $ n $ complex variables with

\noindent frequencies in a finitely generated subgroup $ \Gamma $ o f $ C ˆ{ n }$ i s a function of the form

\ [ \ begin { a l i g n e d } f ( z { 1 } , . . . , z { n } ) = f ( z ) = \sum p {\gamma } ( z ) \exp ( \gamma \cdot z ) , \\ \gamma \ in \Gamma \end{ a l i g n e d }\ ]

\noindent where the sum is finite , the $ p {\gamma }$ are polynomials , \ h f i l l and $ \gamma \cdot z = \gamma 1 ˆ{ z } 1 + . . . + \gamma n ˆ{ z } n . $

\noindent Such a function belongs t o the algebra $ A {\phi } ( C ˆ{ n } ) $ \quad of entire functions $ F $ satisfying t $ h−l i n e { e }$ growth condition :

\ [ \ exists C > 0 \mid F ( z ) \mid \ leq C \exp (C \phi ( z ) ) , \ ]

\ centerline {1 99 1 Mathematics Subject Classification : 32 A 1 5 , 32 A 25 , 34 K 5 , 34 K 35 , 32 S 40 . }

Research of the first author partially supported by the NSF grants DMS − 9000619 and CDR − 880301 2 .

\ centerline { Research of the second author partially supported by the NSF grant DMS − 9000619 . }

\ centerline {The paper is in final form and no version of it will be published elsewhere . }

\ [ [ 77 ] \ ] 78 .. C period A period BERENSTEIN AND A period YGER \noindentwhere the78 weight\quad phi canC .be A taken . BERENSTEIN as bar z bar comma AND A the . YGER Euclidean norm of z comma or even better comma if we choose a system gamma to the power of 1 comma period period period comma gamma to the power of N comma of\noindent Q hyphen linearlywhere independent the weight generators $ \phi of Capital$ can Gamma be taken comma as as $ \mid z \mid , $ the Euclidean norm of $zphi open , $ parenthesis or even z closingbetter parenthesis , = maximum open parenthesis bar Re open parenthesis gamma to the power 78 C . A . BERENSTEIN AND A . YGER of j times z closing parenthesis bar : j = 1 comma period period period comma N closing parenthesis plus log open where the weight φ can be taken as | z |, the Euclidean norm of z, or even better , parenthesis\noindent 1 plusif we bar choose z bar to the a system power of 2 $ closing\gamma parenthesisˆ{ 1 } comma,..., \gamma ˆ{ N } , $ if we choose a system γ1, ..., γN , of − linearly independent generators of Γ, as o fwhere $ Q Re− z denotes$ linearly the real part independent of the complex generatorsQ number z period of .. $ In\Gamma the case that, $ Capital as Gamma subset i R to the power of n comma the exponential polynomials are just the Fourier transforms of distributions φ(z) = max(| <(γj · z) |: j = 1, ..., N) + log(1+ | z |2), \ [ supported\phi ( by finitely z many) = points\max in the lattice( \ minusmid i Capital\Re Gamma( \gamma comma andˆ{ j A sub}\ phicdot i s a subalgebraz ) of\ themid :j=1,...,N)+ \ log ( 1 + \mid zn \mid ˆ{ 2 } Paley endashwhere Wienercomplex number $ z . $ \quad In the case that sometimes . In this case , the exponential polynomials are the Fourier transforms of linear difference $ \Gamma \subset $ called transfer- differential function comma operators characteristic with constant function coefficients comma in symboln( sometimes comma according called transfer to the function area of , $ i R ˆ{ n } , $ the exponential polynomialsR are just the Fourier transforms of distributions mathematicscharacteristic or engineering function involved , symbol in the , problem according closing to the parenthesis area of mathematics period or engineering involved supported by finitely many points in the lattice $ − i \Gamma , $ and $ A {\phi }$ The motivationin the of problemour work ) i . s the study of the solutions of arbitrary systems of i s a subalgebra of the difference hyphenThe differential motivation equations of our work period i s the When study the of dimension the solutions n = of 1 arbitrarycomma the systems sp ectral of difference synthesis theorem open- differential square bracket equations 24 closing . When square the bracket dimension guaranteesn = 1 that, the every sp ectral C to thesynthesis power theorem of infinity [ open 24 ] parenthesis \noindent Paley −− Wiener algebra $ E ˆ{\prime }\widehat {(} R n ) $ \quad of Fourier transforms of distributions of compact or distribution closingguarantees parenthesis that every solutionC∞( orof suchdistribution a system ) solution of such a system support . \quad In this case , the exponential polynomials are the Fourier transforms of open parenthesis( more more generally generally , any comma system any of system convolution of convolution equations equations of compact of compact support support ) b elongs closing to parenthesis linear difference − differential operators with constant coefficients in $ R ˆ{ n } ( $ b elongs the closure of the simplest solutions of the system , namely the exponential polynomial sometimes to the closuresolutions of the simplest that have .. frequencies solutions of in the the system collection comma of common .. namely zeros the exponentialof the transfer functions of called transfer function , characteristic function , symbol , according to the area of polynomial solutionsthe original that equations have frequencies . It is in less the well collection - known of that common the sp zeros ectral of synthesis does not hold mathematics or engineering involved in the problem ) . the transferfor functions arbitrary of the systems original of convolution equations period equations .. It is as less soon well as hyphenn ≥ known2, equivalently that the , not sp ectral synthesisall ideals does in notthe holdPaley for –Wiener arbitrary algebra systems are of lo convolution calizable [ 20equations ] . If an ideal is generated by The motivation of our work i s the study of the solutions of arbitrary systems of as soon as npolynomials greater equal then 2 comma , it has .. b equivalently een proved comma .. not .. all ideals in the Paley endash Wiener algebra are d i f f e r e n c e − differential equations . When the dimension $ n = 1 , $ the sp ectral synthesis lo calizable openby Ehrenpreis square bracket and 20Malgrange closing square , that bracket it is period always .. lo If calizable an ideal is generated [ 1 9 , by 2 1polynomials ] . The then comma theorem [ 24 ] guarantees that every $ C ˆ{\ infty } ( $ or distribution ) solution of such a system it has b een provedonly fairly general criterion to ensure localizability of a finitely generated ideal I is t o verify by Ehrenpreisthat and the Malgrange generators comma form a.. slowly that it decreasing is always lo sequence calizable in .. the open sense square of [ bracket 4 ] . Among 1 9 comma other .. 2 1 closing \noindent ( more generally , any system of convolution equations of compact support ) b elongs square bracket periodrequirements .. The ,only the generators must define a complete intersection . The slowly decreasing to the closure of the simplest \quad solutions of the system , \quad namely the exponential fairly generalcondition criterion is to not ensure t oo easylocalizability t o check of , a finitelyesp ecially generated when the ideal variety I is tV o of common zeros of the polynomial solutions that have frequencies in the collection of common zeros of verify that thegenerators generators is not form discrete a slowly . decreasingThe only general sequence example in the given sense in of [ open4 ] of square a slowly bracket decreasing 4 closing square the transfer functions of the original equations . \quad It is less well − known that the bracket period sequence of exponential polynomials is the following . Let P , ..., P b e polynomials defining sp ectral synthesis does not hold for arbitrary systems1 ofn convolution equations Among othera requirementsdiscrete ( hence comma , finite the ) varietygenerators in C mustn and definek ≤ n, a completethen the sequence intersection of functions period asThe soon slowly as decreasing $ n condition\geq 2 is not , t $oo easy\quad t o checkequivalently comma .. esp , ecially\quad whennot the\quad all ideals in the Paley −− Wiener algebra are lovariety calizable V of common [ 20 zeros ] . of\quad the generatorsIf an idealis not discrete is generated period .. The by only polynomials general then , it has b een proved iz1 izk example given in open square bracketfj(z) 4 = closingPj(e square, ..., e bracket, zk+1, of..., a zn slowly) (1 ≤ decreasingj ≤ n) sequence of exponential (1) polyno- mials\noindent by Ehrenpreis and Malgrange , \quad that it is always lo calizable \quad [ 1 9 , \quad 2 1 ] . \quad The only fairlyis the following generalis slowly period criterion decreasing .. Let P sub , to with 1 ensure comma discrete period localizability ( but period infinite period ) variety comma ofV aof finitely P common sub n bzeros e generated polynomials . defining ideal a $ discrete I $ openi s t parenthesis o henceFor these comma reasons finite , closing in one parenthesis of our previous papers [ 7 ] we considered the case of finitely verifyvariety in that Cgenerated to the the power generators ideals of n of and exponential kform less or a equal polynomials slowly n comma decreasing with then frequencies the sequence sequence in a of group functions in the Γ of sense rank exactly of [ 4 ] . n AmongEquation: other openn and requirements parenthesis assumed that 1 closing ,V thewas parenthesis generatorsdiscrete . .. f Even sub must j when open define Γ parenthesis = i aZ , completewe z could closing not intersection parenthesis find a general = P . sub j open parenthesisThe slowly e tocriterion thedecreasing power for of localizability i zcondition sub 1 comma of the is idealsperiod not generated periodt oo easyperiod by such t comma o exponential check e to the , polynomials\ powerquad ofesp i z . sub ecially Part k comma of when z sub the k plusvariety 1 comma $ periodthe V $problem period of common i period s of an comma zerosarithmetic z of sub nature the n closing generators , namely parenthesis lo calizability is open not parenthesis may discrete depend 1 less . not\ orquad only equal onThe j the less only or equal general n closingexample parenthesis givengeometry in [ of 4V ]and of Γ, abut slowly also on decreasing the diophantine sequence approximations of exponential of the coefficients polynomials of the is the following . \quad Let $ P { 1 } ,...,P { n }$ b e polynomials defining a discrete ( hence , finite ) is slowly decreasinggenerators comma of I. withFor example discrete , open the ideal parenthesis generated but by infinite cos (z1 closing), cos ( parenthesisz2), z2 − αz1 varietyis lo calizable V of common zeros periodvariety inif and $C only ˆ{ ifnα}i$ s not and a Liouville $ k number\ leq .n As we , pointed $ then outthe in [ sequence 8 ] , there of i s afunctions deep For these reasonsrelationship comma in one of our previous papers open square bracket 7 closing square bracket we considered the\ begin case{ ofa l i g n ∗} \ tagfinitely∗{$ generated( 1 ideals ) $} off exponential{ j } polynomials( z ) with = frequencies P { j in} a group( e ˆ{ i z { 1 }} ,. .Capital . ,Gamma e ˆ of{ ranki exactly z { nk and}} assumed, z that{ Vk was + discrete 1 } period, .. . Even . when . Capital , Gamma z { n = i} Z to) the power( 1 of n\ commaleq wej \ leq n ) \endcould{ a l noti g n find∗} a general criterion for localizability of the ideals generated by such exponential polynomials period .. Part of the problem i s of an arithmetic nature comma namely \noindentlo calizabilityis may slowly depend decreasing not only on the , with geometry discrete of V and ( Capital but infinite Gamma comma ) variety but also on $ Vthe $ of common zeros . diophantine approximations of the coefficients of the generators of I period For example comma Forthe these ideal generated reasons by , cos in open one parenthesis of our previous z sub 1 closing papers parenthesis [ 7 ] comma we considered cos open parenthesis the case z sub of 2 closing parenthesisfinitely comma generated z sub 2minus ideals alpha of z exponential sub 1 is lo calizable polynomials if and onlyif with alpha frequencies i s in a group $not\Gamma a Liouville$ number of rank period exactly .. As we $ pointed n $ out and in assumed.. open square that bracket $ V 8 $closing was square discrete bracket . comma\quad ..Even there when i$ s\ aGamma deep relationship= i Z ˆ{ n } , $ we could not find a general criterion for localizability of the ideals generated by such exponential polynomials . \quad Part of the problem i s of an arithmetic nature , namely lo calizability may depend not only on the geometry of $ V $ and $ \Gamma , $ but also on the diophantine approximations of the coefficients of the generators of $ I . $ For example , the ideal generated by cos $ ( z { 1 } ) , $ cos $ ( z { 2 } ) , z { 2 } − \alpha z { 1 }$ is lo calizable if and only if $ \alpha $ i s not a Liouville number . \quad As we pointed out in \quad [ 8 ] , \quad there i s a deep relationship EXPONENTIAL POLYNOMIALS .. 79 \ hspacebetween∗{\ thef i lo l l calizability}EXPONENTIAL i ssue and POLYNOMIALS the following\ conjquad ecture79 of Ehrenpreis on the zeros of exponential polynomials of a single variable with algebraic coefficients \noindentand frequenciesbetween period the lo calizability i ssue and the following conj ecture of Ehrenpreis on the zerosConjecture of exponential 0 period 1 period polynomials .. Let a sub 1 of comma a single period period variable period with comma algebraic a sub m in Q coefficients open square bracket z EXPONENTIAL POLYNOMIALS 79 closingand frequencies square bracket .. . be non hyphen zero comma alpha sub 1 comma period period period comma alpha sub m .. be between the lo calizability i ssue and the following conj ecture of Ehrenpreis on the zeros of distinct exponential polynomials of a single variable with algebraic coefficients and frequencies . \ hspacenumbers∗{\ inf Q i l intersectionl } Conjecture of R comma 0 . 1 .. . and\quad f be theLet exponential $ a { polynomial1 } , of . a single . variable . , given a { m }\ in Conjecture 0 . 1 . Let a , ..., a ∈ [z] be non - zero , α , ..., α be Qby [ z ] $ \quad be non − zero $1 , \malphaQ { 1 } ,...,1 m \alpha { m }$ distinct \quadLinebe 1 m d Line i s t i2 n f c open t parenthesis z closing parenthesis = sum a sub j open parenthesis z closing parenthesis e to the numbers in T , and f be the exponential polynomial of a single variable given by power of i alpha sub j z periodQ LineR 3 j = 1 \noindent numbers in $ Q \bigcap R , $ \quad and $ f $ be the exponential polynomial of a single variable given In that case comma there exist constants epsilon greater 0 comma Nm greater equal 0 such that if z sub 1 comma z sub 2by are two different X iαj z zeros of f then f(z) = aj(z)e . \ [ \ begin { a l i g n e d } m \\ bar z sub 1 minus z sub 2 bar greater equal open parenthesis barj = z1 sub 1 bar epsilon plus bar z sub 2 bar closing parenthesisf ( to z the power ) = of N\ periodsum a { j } ( z ) e ˆ{ i \alpha { j } z } . \\ jIn our = more 1 In recent\end that{ paper casea l i g nopen, ethere d }\ square exist] bracket constants 14 closingε > 0,N square≥ 0 such bracket that .. if wez consider1, z2 are a two situation different that zeros is fairly different from that ofof openf then square bracket 7 closing square bracket period .. Namely comma the group Capital Gamma has very low rank comma either one or two comma and N \noindentthe varietyIn V might that not case b e discrete , there or existcomplete| z1 − constantsz intersection2 |≥ (| z1 period| $ε+\|varepsilonz Using2 |) . a convenient> 0 , N \geq 0 $ such that if $ z { 1 } , z { 2 }$ are two different generalization ofIn the our Weyl more algebra recent papercomma [ 14we ] have we obtained consider some a situation results that that is are fairly very different from that zeros of $ f $ then simple to stateof [ period 7 ] . .. ForNamely instance , the comma group if Γ rank hasvery open low parenthesis rank , either Capital one Gamma or two closing , and theparenthesis variety = 1 comma any system of exponentialV might not polyno b e discrete hyphen or complete intersection . Using a convenient generalization of the \ [ \mid z { 1 } − z { 2 }\mid \geq ( \mid z { 1 }\mid \ varepsilon { + } mials definingWeyl a complete algebra , intersection we have obtained generates some a lo results calizable that ideal are veryin the simple space to state . For instance \mid z { 2 }\mid ) ˆ{ N } . \ ] A sub phi period, if rank .. Another (Γ) = 1, exampleany system of localizability of exponential occurs polyno when - mials the generators defining a are complete of the type intersection open parenthesisgenerates 1 closing a lo calizable parenthesis ideal and in thedefine space a nonAφ. hyphenAnother discrete example complete of localizability intersection occurs period when .. We have also studied prob hyphenthe generators are of the type In our more recent paper [ 14 ] \quad we consider a situation that is fairly different lems related( t 1 o ) global and define versions a non of the - discrete Nullstellensatz complete and intersection of the Brian . c-cedilla We have on also endash studied Skoda prob - lems from that of [ 7 ] . \quad Namely , the group $ \Gamma $ has very low rank , either one or two , and theorem commarelated which t o could global b versions e useful when of the solving Nullstellensatz the ubiquitous and of Bezout the Brian identity c − forcedilla on – Skoda the variety $ V $ might not b e discrete or complete intersection . Using a convenient exponentialtheorem polynomials , which without could common b e useful zeros when period solving .. The the solutionubiquitous of Bezoutthe Ehrenpreis identity for exponential generalization of the Weyl algebra , we have obtained some results that are very conj ecture mentionedpolynomials earlier without comma common is precisely zeros . equivalent The solution t o solving of the Ehrenpreisin general the conj Be ecture hyphen mentioned simple to state . \quad For instance , if rank $ ( \Gamma ) = 1 , $ any system of exponential polyno − zout identityearlier period , is precisely equivalent t o solving in general the Be - zout identity . mials defining a complete intersection generates a lo calizable ideal in the space The leitmotiv ofThe our leitmotiv approach of i s our to approachrelate the i division s to relate problems the division implicit problems in implicit in the previous $ A {\phi } . $ \quad Another example of localizability occurs when the generators are of the type the previousquestions questions , comma t o the tstudy o the study of the of analytic the analytic continuation continuation , as commaa function as a of functionλ1, ..., λ ofm, of the lambda subdistribution 1 comma period period period comma lambda sub m comma of the distribution \noindentz mapsto-arrowright( 1 ) and bar define f 1 open a parenthesis non − discrete z closing parenthesis complete bar intersection to the power of . lambda\quad subWe 1 have period also period studied prob − lems related t o global versions of the Nullstellensatzλ1 λm and of the Brian $ c−c e d i l l a $ period bar f sub m open parenthesis z closing parenthesisz 7→| f1(z) bar| ... to| thefm( powerz) | of, lambda sub m comma on for−− exponentialSkoda polynomials f sub j period .. In fact comma the residues of this distribution hyphen valued theoremmeromorphic , whichfor functions exponential could give polynomials b the e information usefulfj. when neededIn fact solving t , o the solve residues the the Bezout ubiquitous of this equation distribution Bezout - valued identity meromor- for exponentialand other divisionphic polynomials functions problems give period without the information.. This common idea originatedneeded zeros t o in solve . our\quad the previous BezoutThe work solution equation about and of other the division Ehrenpreis conjresidue ecture currentsproblems mentioned open . square This earlier bracket idea originated 2 closing , is preciselyin square our previous bracket equivalent and work their about applications residue t o solving currents t o the [in effective 2 ] andgeneral their solvability the of Be the− polyzout hyphen identityapplications . t o the effective solvability of the poly - nomial membership problem [ 1 0 , 1 nomial membership1 ] . The problem reader .. can open consult square our bracket forthcoming 1 0 comma monograph .. 1 1 closing [ 3 ] , forsquare an overview bracket period of the ..ideas The reader can consultThe leitmotiv our forthcomingwe use of . our The approach theory of iD− smodules to relate , as introduced the division by J . Bernstein problems [ 16 implicit ] , was precisely in themonograph previousformulated open questions square to bracket obtain , t3 an oclosing explicit the square study bracket of the comma analytic for an overview continuation of the ideas , as we a use function period .. The of λ theory$ \lambda of D hyphenform{ modules1 of the} analytic,..., comma continuation in λ of\ thelambda distribution{ m |}P (z),| $when ofP theis a distributionpoly - nomial . as introducedBernstein by J period ’ s results Bernstein were open extended square by bracket Bjo ¨ rk 16 t o closing the holomorphic square bracket setting comma in [ 1 was 7 ] .precisely The formulated to\ [ obtain z \ anmapsto explicitmain point\mid used inf Bernstein 1 ( ’ s work zi ) s that\mid oneˆ deals{\lambda with holonomic{ 1 }}D− modules... , the \mid f form{ m of} the( analyticnew z difficulty continuation ) that\mid arises inˆ{\ lambda forlambda exponential of the{ distributionm polynomials}} , bar\ ] i P s thatopen the parenthesis z closing parenthesis bar to the power of lambdaD− whenmodules P is a one poly needs hyphen to consider are not holonomic . nomial period .. Bernstein quoteright s results were extended by Bj o-dieresis rk t o the holomorphic setting in \noindentopen squarefor bracket exponential 1 7 closing square polynomials bracket period $ f .. The{ j main} . point $ used\quad in BernsteinIn fact quoteright , the residues s work i s that of this distribution − valued onemeromorphic deals with holonomic functions give the information needed t o solve the Bezout equation andD hyphen other modules division comma problems the new difficulty . \quad thatThis arises idea for exponential originated polynomials in our i previous s that the work about residueD hyphen currents modules one [ needs 2 ] and to consider their are applications not holonomic period t o the effective solvability of the poly − nomial membership problem \quad [ 1 0 , \quad 1 1 ] . \quad The reader can consult our forthcoming monograph [ 3 ] , for an overview of the ideas we use . \quad The theory of $ D − $ modules , as introduced by J . Bernstein [ 16 ] , was precisely formulated to obtain an explicit

\noindent form of the analytic continuation in $ \lambda $ of the distribution $ \mid P ( z ) \mid ˆ{\lambda }$ when $P$ is a poly − nomial . \quad Bernstein ’ s results were extended by Bj $ \ddot{o} $ rk t o the holomorphic setting in [ 1 7 ] . \quad The main point used in Bernstein ’ s work i s that one deals with holonomic $ D − $ modules , the new difficulty that arises for exponential polynomials i s that the

\noindent $ D − $ modules one needs to consider are not holonomic . 80 .. C period A period BERENSTEIN AND A period YGER \noindent1 period ..80 A generalization\quad C . A of the. BERENSTEIN Weyl algebra AND period A .. . The YGER ideas we present in this sec hyphen tion are clearly related t o those about the Weyl algebra found in open square bracket 1 7 comma Chapter 1 closing square1 . \quad bracketA comma generalization of the Weyl algebra . \quad The ideas we present in this sec − tionto which are we clearly refer for furtherrelated developments t o those period about the Weyl algebra found in [ 1 7 , Chapter 1 ] , 80 C . A . BERENSTEIN AND A . YGER toWe which .. denote we by refer N the for.. set further .. of non hyphen developments negative integers . period .. For .. an index .. alpha in N to the power of 1 . A generalization of the Weyl algebra . The ideas we present in this sec n comma - tion are clearly related t o those about the Weyl algebra found in [ 1 7 , Chapter 1 ] , to We it\quad s lengthdenote .. bar alpha by $N$bar = alpha the sub\quad 1 pluss period e t \quad periodo periodf non plus− negative alpha sub n integers period .. We . \ alsoquad letFor K b e\quad a an index \quad which we refer for further developments . field$ \alpha of characteristic\ in zeroN ˆ comma{ n } , $ We denote by the set of non - negative integers . For an index α ∈ n, i tn and s length m two positive\quad integers$ \mid commaN \ wealpha define an\mid extension= E sub\alpha n comma{ m1 open} + parenthesis . . K closingN . parenthesis+ \alpha { n } it s length | α | = α + ... + α . We also let b e a field of characteristic zero , n and of. the $ Weyl\quad algeWe hyphen also let $K$1 b e an field of characteristicK zero , m two positive integers , we define an extension E ( ) of the Weyl alge - $bra n A $ sub and n open $m$ parenthesis two K positive closing parenthesis integers period , we ..n,m defineIt iK s an algebra an extension of operators $ acting E { onn the , algebra m } of bra A ( ). It i s an algebra of operators acting on the algebra of polynomials in n + m polynomials( K )$ in ofn theWeylalgeK − variables over as follows . n plus m variables over KK as follows period Consider the polynomial algebra [x , ..., x , y1, ..., ym] and derivations D , ..., D \noindentConsider ..bra the .. $ polynomial A { n } .. algebra( K .. K ) open . square $ K\ bracketquad1 It xn sub i s1 comma an algebra period period of operators period1 comma actingn x sub on the algebra of polynomials in on this algebra such that n comma$n y + 1 comma m$ period variables period period over comma $K$ y m as closing follows square bracket. .. and .. derivations

D sub 1 comma period period period comma D subD nix onj = thisδij algebra(i, j = such 1, ..., that n), ConsiderLine 1 D\ subquad i xthe sub j\ =quad deltapolynomial sub ij open parenthesis\quad algebra i comma j\ =quad 1 comma$ K period [ period x { period1 } comma,.. n closing D y = δ y (i = 1, ..., n; j = 1, ..., m). parenthesis. , x comma{ n Line} ,y1,...,ym]$ 2 D sub i y sub ji =j deltaij subj ij y sub j open parenthesis i\ =quad 1 commaand period\quad periodderivations period $ D { 1 } ,...,D { n }$ on this algebra such that comma n semicolonThe algebraj = 1 commaEn,m( periodK) i s theperiod algebra period of comma operators mclosing on K[x1 parenthesis, ..., xn, y1, ..., period ym] gen - erated by The algebraX E1, sub ..., X nn comma,Y1, ..., mYm open,D1, ...,parenthesis Dn, where KX closingi( resp parenthesis.Yj) i s thei s the operator algebra of of multiplication operators on byK open square \ [ \ begin { a l i g n e d } D { i } x { j } = \ delta { i j } ( i , j = 1 , . bracket x sub 1x commai( resp period.yj). It period i s a Lie period algebra comma , with x sub the n usual comma definition y 1 comma of the period Lie periodbracket period [ . , . comma ] in t y m closing square. . bracket , generms n hyphen of ) the composition , \\ of operators , i . e . , Derated{ i by} X suby 1{ commaj } period= \ perioddelta period{ i j comma} y X sub{ j n} comma( Y i sub = 1 comma 1 period , .period . period . comma , Yn;j=1,...,m). sub m comma D sub 1 comma period period period[P,Q comma] = P ◦ DQ sub− Q n\◦end commaP. { a l where i g n e d X}\ sub] i open parenthesis resp period Y sub j closing parenthesis i s the operator The Lie bracket satisfies the following commutator relations of multiplication by x sub i open parenthesis resp period y sub j closing parenthesis period It i s a Lie algebra comma with\noindent the usualThe definition algebra of $ E { n , m } ( K ) $ i s the algebra of operators on = [Yi,Yj] = [Xi,Yj] = [Di,Dj] = 0; $ Kthe Lie [ bracket x { open1 } square, bracket. . period . comma , x period{ n closing} , square y 1bracket , in .t erms . of the . composition , y m of [X ,D ] = −δ ;[Y ,D ] = −δ Y . operators] $ gen comma− i period e period comma i j ij i j ij i eratedopen square byWe bracket $ X note{ P that1 comma} for m,...,X Q closing= 0 square our algebra bracket coincides = P{ circ withn Q} minus the,Y Weyl Q circ algebra{ P1 period} .,..., Every element Y { m } ,D { 1 } ,...,D { n } , $ where $ X { i } ( $ resp The Lie bracketP of satisfiesEn,m(K) the can following b e written commutator in a unique relations way as a finite sum $ .Line Y 1 open{ j square} ) bracket $ i X s sub the i comma operator X sub j closing square bracket = open square bracket Y sub i comma Y subof j closing multiplication square bracket by = open $ x square{ i bracket} ( $ X sub resp i comma $ . Y sub y j closing{ j } square) bracket . $ It = open i s square a Lie bracket algebra , with the usual definition of the Lie bracket [ . , . ] in t erms of theX compositionα β γ of operators , i . e . , D sub i comma D sub j closing square bracket =P 0= semicoloncα,β,γ LineX 2Y openD square, bracket X sub i comma(2) D sub j closing square bracket = minus delta sub ij semicolon open square bracket Y subα, β, i γ comma D sub j closing square bracket = minus delta\ [ [ sub P ij Y sub , i period Q ] = P \ circ Q − Q \ circ P. \ ] n m We note thatcα,β,γ for m =∈ 0 ourK,algebra α, γ ∈ coincidesN , β with∈ theN Weyl. algebraThe integer period max .. Every (| α | + | β | + | γ | element P of: E subc n comma6= m0) open is denoted parenthesis deg P. K closingIt i s parenthesisconvenient t can o introduce b e written the in operators a unique wayad(Q as) a finite sum \noindent The Lieα,β,γ bracket satisfies the following commutator relations Equation: openacting parenthesis on En,m(K 2) closing by ad( parenthesisQ)(P ) := [P,Q .. P]. = sumOne c has sub the alpha following comma simple beta comma calculus gamma rules . X to the power of alpha Y to the power of betaLemma D to the1 . power 1 . For of any gamma integers commaa, balpha≥ 0, 1 comma≤ k ≤ betan, we comma have gamma \ [ \cbegin sub alpha{ a l i comma g n e d } beta[X comma{ gammai } ,X in K comma{ j alpha} ] comma = gamma [ Y in N{ toi the} power,Y of n{ commaj } beta] in = N to[X the power{ i of} m period,Y .. The{ j integer} ] max = open [ parenthesis D { i bar} alpha,D bar plus{ j bar} beta] bar = plus 0 bar ; gamma\\ bar : c [X { i } ,D { j } ] = − \ delta { i j } ;[Ya−1 b { i a} b ,D { j } sub alpha comma beta comma gamma negationslash-equal 0 closing parenthesis = aXk Yk + bXk Yk . ] = − \ delta { i j } Y { i } . \end{ a l i g n e d }\ ] is denoted deg P period .. It i s convenient t o introduce theM operators ad open parenthesisN Q closing parenthesis .. X k 0 X l 0 acting on Corollary 1.1. LetP (X,Y ) = X1 Pk(X ,Y ) = Y1 Ql(X,Y ), E sub n comma m open parenthesis K closing parenthesis by adk=0 open parenthesis Q closingl=0 parenthesis open parenthesis \noindent We note that for $m = 0 $ our algebra coincides with the Weyl algebra . \quad Every P closing parenthesis : = open square bracket P commawhereX Q closing= square(X ,X0 bracket),Y − line period= (Y ..,Y One0). has T henthe following simple element $P$ of $E { n , m } ( K )1 $ can b e written1 in a unique way as a finite sum calculus rules period l=0 k=0 X X Lemma 1 period[D 1 period,P ] = ForY anyl(∂Ql integers+ lQl a comma) = X bM greaterY ∂PM equal+ 0 commaXk{(k 1+ less 1)P or equal+ Y k∂P lessk } or. equal n comma \ begin { a l i g n ∗} 1 1 ∂ X1 1 1 ∂ Y1 1 k+1 1 ∂ Y1 we have N M−1 \ tagLine∗{$ 1 open ( square 2 ) bracket $} P D sub = k comma\sum X subc k{\ toalpha the power of, a Y\ subbeta k to the, power\gamma of b closing} X square ˆ{\alpha bracket } Y ˆ{\beta } D ˆ{\gamma } , \\\alpha , \beta , \gamma = aX sub k to the power of a minusThese 1 calculus Y sub k rules to the are power necessary of b plus to study bX sub the k natural to the powerfiltration of aEv Y, defined sub k to the power of\end b period{ a l i g Linen ∗} 2 Corollary 1 period 1 period Let P open parenthesis X comma Y closing parenthesis = sum sub k = 0 to the power of M X sub 1 to the power of k P sub k open parenthesis X to the power of prime comma Y closing parenthesis\noindent = sum$ c sub{\ l =alpha 0 to the power, \ ofbeta N Y sub, 1 to\ thegamma poweronE}\n,m of l(K Qin)by l openK, parenthesis\alpha X comma, Y to\ thegamma power \ in N ˆ{ n } , \beta \ in N ˆ{ m } . $ \quad The integer max $ ( \mid \alpha of prime closing parenthesis comma Line 3 whereEv := X{P =∈ openEn,m parenthesis(K) : deg P X≤ subv}. 1 comma X to the power of prime closing parenthesis\mid + comma\mid Y-line\beta = open parenthesis\mid + Y sub\mid 1 comma\gamma Y to the power\mid of prime: c closing{\alpha parenthesis, period\beta Then Line, 4\gamma open square}\ bracketnot= D 0 sub 1 ) comma $ P closing square bracket = sum from l = 0 to N Y sub 1 to the power of l parenleftbiggis denoted sub deg partialdiff $P sub . X $ sub\ 1quad to theIt power i s of convenient partialdiff Q l t plus o lQ introduce l parenrightbigg the operators = X sub 1 to the $ ad power of( M QY sub ) 1 $partialdiff\quad suba c partialdiff t i n g on sub Y sub 1 to the power of P sub M plus sum from k = 0 to M minus 1 X sub 1 to the$ E power{ n of k braceleftbigg, m } (K)$by$ad open parenthesis k plus 1 closing (Q) parenthesis P (P)sub k plus 1 plus : Y sub =[ 1 partialdiff P sub , Qpartialdiff ] sub . $ Y sub\quad 1 toOne the power has of the P sub following k bracerightbigg simple period calculus rules . These calculus rules are necessary to study the natural filtration E sub v comma defined \ centerlineon E sub n comma{Lemma1 m open . 1 parenthesis . For any K closing integers parenthesis $ a by E , sub bv : =\ opengeq brace0 P in , E sub 1 n comma\ leq mk open parenthesis\ leq n K closing , $ parenthesiswe have } : degree P less or equal v closing brace period \ [ \ begin { a l i g n e d } [D { k } , X ˆ{ a } { k } Y ˆ{ b } { k } ] = aX ˆ{ a − 1 } { k } Y ˆ{ b } { k } + bX ˆ{ a } { k } Y ˆ{ b } { k } . \\ Corollary 1 . 1 . Let P ( X , Y ) = \sum ˆ{ M } { k = 0 } X ˆ{ k } { 1 } P { k } ( X ˆ{\prime } , Y ) = \sum ˆ{ N } { l = 0 } Y ˆ{ l } { 1 } Q l ( X , Y ˆ{\prime } ), \\ where X = ( X { 1 } , X ˆ{\prime } ),Y−l i n e = ( Y { 1 } , Y ˆ{\prime } ) . Then \\ [D { 1 } , P ] = \sum ˆ{ l = 0 } { N } Y ˆ{ l } { 1 } ( {\ partial }ˆ{\ partial Q l } { X { 1 }} + lQ l ) = X ˆ{ M } { 1 } Y { 1 }\ partial {\ partial }ˆ{ P { M }} { Y { 1 }} + \sum ˆ{ k = 0 } { M − 1 } X ˆ{ k } { 1 }\{ ( k + 1 ) P { k + 1 } + Y { 1 }\ partial {\ partial }ˆ{ P { k }} { Y { 1 }}\} . \end{ a l i g n e d }\ ]

\ hspace ∗{\ f i l l }These calculus rules are necessary to study the natural filtration $ E { v } , $ d e f i n e d

\ begin { a l i g n ∗} on E { n , m } ( K ) by \\ E { v } : = \{ P \ in E { n , m } (K): \deg P \ leq v \} . \end{ a l i g n ∗} EXPONENTIAL POLYNOMIALS .. 81 \ hspaceIt is a∗{\ K hyphenf i l l }EXPONENTIAL vector space of POLYNOMIALS dimension parenleftbig\quad to81 the power of 2 n plus sub v to the power of m plus v parenrightbig thickapprox v to the power of 2 n plus m period As usual open square bracket 1 7 closing square bracket comma\noindent one introI t hyphen i s a $ K − $ vector space of dimension $ ( ˆ{ 2 n + }ˆ{ m + v } { v } ) duces\ thickapprox the graded algebra v ˆ{ gr2 open n parenthesis + m E} sub. n$ comma As usual m open [ 1parenthesis 7 ] , one K closing intro parenthesis− closing EXPONENTIAL POLYNOMIALS 81 parenthesis by It is a − vector space of dimension (2n+m+v) v2n+m. As usual [ 1 7 ] , one intro - \noindentgr open parenthesisducesK the E sub graded n comma algebra m open gr parenthesis $ (v K E≈ closing{ n parenthesis , m } closing( parenthesis K ) : ) = $ E sub by 0 oplus duces the graded algebra gr (E ( )) by E sub 1 slash E sub 0 oplus period period periodn,m K \ [The gr above ( lemma E { cann be used, m t o} show( this K algebra ) i s ) commutative : = period E { ..0 Moreover}\oplus comma E { 1 } /E { 0 } gr(E ( )) := E ⊕ E /E ⊕ ... \oplusgr open parenthesis... E sub\ ] n comma m open parenthesisn,m K K closing0 1 parenthesis0 closing parenthesis is i somorphic t o a polynomial ringThe in 2 above n plus lemma m variables can be period used t o show this algebra i s commutative . Moreover , gr Let M be a open(En,m parenthesis(K)) is i somorphic left closing t o parenthesis a polynomial E sub ring n in comma 2n + m mvariables open parenthesis . K closing parenthesis hyphen \noindent The above lemma can be used t o show this algebra i s commutative . \quad Moreover , module and CapitalLet GammaM be a sub ( left v a )E filtrationn,m(K)− ofmodule M comma and iΓ periodv a filtration e period of M, commai . e .an , an increasing increasing family gr $ ( E { n , m } ( K ) ) $ is i somorphic t oa polynomial ring in $2 family of finiteof finite dimensional dimensional K hyphenK− vector vector spaces spaces Γ Capitalv such that Gamma sub v such that nLine + 1 m$open parenthesis variables i closing . parenthesis union of sub v greater equal 0 Capital Gamma sub v = M semicolon [ Line 2 open parenthesis ii closing parenthesis X sub i Capital Gamma(i) sub vΓ subsetv = M equal; Capital Gamma sub v plus 1 commaLet $M$ Y sub i Capitalbea(left Gamma sub $) v subset E { equaln Capital , m Gamma} (K) subv v≥0 plus 1 comma− $ module and D sub and i Capital $ \Gamma Gamma{ v }$ a filtration of $M , $ i . e . , an increasing sub v subset equal Capital Gamma sub(ii) vX plusiΓv ⊆ 1 periodΓv+1,YiΓv ⊆ Γv+1, andDiΓv ⊆ Γv+1. familyLet Capital of finite Gamma open dimensional parenthesis v $ closing K − parenthesis$ vector : = spaces Capital Gamma $ \Gamma sub v{ slashv } Capital$ such Gamma that sub v minus 1 and defineLet Γ(grv open) := Γ parenthesisv/Γv−1 and M define closing gr parenthesis (M) by by \ [ \grbegin open{ parenthesisa l i g n e d } M( closing i parenthesis ) \bigcup : = Capital{ v Gamma\geq sub 0 oplus0 }\ CapitalGamma Gamma{ v sub} 1= slash M Capital ; Gamma\\ sub( 0 oplus i i period ) period X { periodi }\ =Gamma Capitalgr(M) Gamma{ :=v Γ0}\⊕ openΓ1subseteq/Γ parenthesis0 ⊕ ... = Γ(0) 0\Gamma⊕ closingΓ(1) ⊕ parenthesis{... v + oplus 1 Capital} ,Y Gamma{ i open} \Gamma { v }\subseteq \Gamma { v + 1 } , and D { i }\Gamma { v }\subseteq parenthesis 1 closingDue t parenthesis o property oplus( ii ) , period this graded period module period is a module over gr (E ( )). One says the \Gamma { v + 1 } . \end{ a l i g n e d }\ ] n,m K Due t o propertyfiltration open is parenthesis a good filtration ii closingif parenthesis gr (M) i s ofcomma finite this typ graded e over modulegr (En,m is(K a module)). For over instance gr open , parenthesis E sub n commaif mM openi s finitely parenthesis generated K closing over E parenthesisn,m(K) by closinga1, ..., parenthesis ar and we period choose .. Γ Onev := saysEva1 +...+Evar, the filtrationthen is a wegood have filtration a good if filtration gr open parenthesis . As in [ M1 7 closing , Lemma parenthesis 3 . 4 ] , i s of finite typ e over gr open parenthesis \noindent Let $ \Gamma ( v ) : = \Gamma { v } / \Gamma { v − 1 }$ E sub n commaone m open can prove parenthesis the following K closing lemma parenthesis . closing parenthesis period .. For anddefine gr $( M )$ by instance commaLemma .. if M i s1 finitely . 2 . Let generated(Γv) overv, (Ω Ev)v subbe n comma two filtrations m open parenthesis of an En,m K closing(K) module parenthesis .. by a sub 1 comma periodM, and period assume period that comma(Γv)v a subis a r good .. and filtration we choose . Then there is an integer w such that \ [Capital gr ( Gamma M sub ) v : : = E = sub v\Gamma a sub 1 plus{ 0 period}\oplus period period\Gamma plus E{ sub1 v} a sub/ r comma\Gamma then{ we0 have}\ aoplus good. filtration . . period = \ ..Gamma As in open( square 0 bracket ) \ 1oplus 7 comma Lemma\Gamma 3 period( 4 1 closing ) square\oplus bracket... comma \ ] one can prove the following lemma period Γv ⊆ Ωv+wforallv ≥ 0. Lemma .. 1 period 2 period .. Let .. open parenthesis Capital Gamma sub v closing parenthesis sub v comma open \noindent Due t o property ( ii ) , this graded module is a module over gr $ ( E { n parenthesis CapitalIf Omega gr (M) sub i s v closing of finite parenthesis type over sub gr v (E ..n,m be(K ..)), twothere filtrations i s a .. Hilbert of an E polynomial sub n comma m open parenthesis, m } K( closingH K∈ Q parenthesis[t )] such ) that .. . for module $ all\vquad .. M1 commaOne says theand filtration assume that open is a parenthesis good filtration Capital Gamma if gr sub $v closing ( M parenthesis ) $ i sub s ofv .. finite is a good typ filtration e over period gr .. $ ( E { n , m } ( K ) ) . $ \quad For Then there is an integer w such that H(v) = dimK Γv i n s t a n c e , \quad if $M$ i s finitely generated over $E { n , m } ( K ) $ Capital Gamma( see sub [ 1 7v ,subset Theorem equal 3 Capital. 1 ] ) . Omega As a sub consequence v plus w for of the all v last greater lemma equal , the 0period degree and the \quadIf grby open$ parenthesis a { 1 M} closing, . parenthesis . . .. i s , .. of a finite{ typer }$ .. over\quad gr openand parenthesis we choose E sub n comma m open $ \Gamma leading{ v } coefficient: = of H E do{ notv depend} a on{ the1 } choice+ of the . good . filtration . + (Γv E)v. The{ v degree} ad { r } , $ parenthesis K closingof H is parenthesis called the dimension closing parenthesisd(M) comma of gr (M ..) there and i thes ..multiplicity a Hilbert .. polynomiale(M) of gr (M) i s thenH in we Q openhave square a good bracket filtration t closing square . \quad bracketAs such in that [ 17 for all ,Lemma3 v gg 1 . 4 ] , the leading t erm of H t imes d!. In the case m = 0, i . e . , for the Weyl algebra An(K) one H open parenthesis v closing parenthesis = dimension sub K Capital Gamma sub v \noindent onehas canthe fundamental prove the theorem following of J . Bernstein lemma . that asserts that , for any non - trivial An(K)− open parenthesismodule seeM openso that square gr ( bracketM) is of 1 finite 7 comma typ e Theorem , 3 period 1 closing square bracket closing parenthesis period .. As a consequence of the last lemma comma the degree and the Lemma \quad 1 . 2 . \quad Let \quad $ ( \Gamma { v } ) { v } ,( \Omega { v } leading coefficient of H do not depend on the choiced( ofM the) ≥ goodn. filtration open parenthesis Capital Gamma sub v closing) { v parenthesis}$ \quad subbe v period\quad two filtrations \quad o f an $ E { n , m } ( K ) $ \quad module \quad $ M , $ The degree dAn ofA Hn is(K called)− module the dimensionM such that d opend(M parenthesis) = n i s said M closing to b e parenthesisholonomic ... of gr open parenthesis M closing parenthesisand assume .. and that theOne multiplicity of $ the ( applications\Gamma { ofv the} concept) { ofv } holonomic$ \quad modulesis a goodi s the filtration existence of the . \quad Then there is an integer $ we open $ such parenthesisBernstein that M closing – Sato parenthesisfunctional equations .. of gr open[ 1 7parenthesis , 25 , M 23 closing ] , i .parenthesis e . , given i s polynomialsthe leading t erm of H t imes d ! periodf ..1, In ..., the fq in caseK[x m1, = ..., 0 x comman] there i period are differential e period comma operators for Qj in An(K[λ]), with \ begin { a l i g n ∗} the Weyl algebraλ = ( Aλ1 sub, ..., n λq open), and parenthesis a non - zero K closing polynomial parenthesisb ∈ K[ oneλ] such has thethat fundamental the formal rela theorem - tions of J period Bernstein \Gamma { v }\subseteq \Omega { v + w } f o r a l l v \geq 0 . that λ +1 \end{ a l i g n ∗} λ1 j λq λ1 λq asserts that comma for any nonQ hyphenj(f1 ...f trivialj ...f A sub) =n openb(λ)f1 parenthesis...f (j K= closing 1, ..., q) parenthesis hyphen module M so that gr open parenthesishold . M closing parenthesis is of finite typ e comma I fd gr open $ parenthesis ( M M ) closing $ \quad parenthesisi s \ greaterquad of equal finite n period type \quad over gr $ ( E { n , m } (An K A sub ) n open ) parenthesis , $ \quad K closingther parenthesis e i s \quad hyphena module H i l b e M r t such\quad that dpolynomial open parenthesis M closing parenthesis = n$ i H s said\ toin b e holonomicQ [ period t ]$ suchthatforall $v \gg 1 $ One of the applications of the concept of holonomic modules i s the existence \ [of H the ( Bernstein v endash ) = Sato\dim functional{ K equations}\Gamma open square{ v }\ bracket] 1 7 comma .. 25 comma .. 23 closing square bracket comma .. i period e period comma .. given polynomials f 1 comma period period period comma f sub q in K open square bracket x sub 1 comma period period period comma x\noindent sub n closing( square see [ bracket 17 , .. Theorem3 there are differential . 1 ] ) operators . \quad Q subAs j a .. consequence in A sub n open of parenthesis the last K lemmaopen square , the degree and the bracketleading lambda coefficient closing square of bracket $ H closing $ do parenthesis not depend comma on with the choice of the good filtration $ ( \Gammalambda{ = openv } parenthesis) { v } lambda. $ sub 1 comma period period period comma lambda sub q closing parenthesis comma andThedegree a non hyphen $d$ zero polynomial of $H$ b in K is open called square bracket the dimension lambda closing $d square ( bracket M such ) $ that\quad the formalo f gr rela hyphen$ ( M ) $ \quad and the multiplicity $tions e ( M ) $ \quad of gr $( M )$ i s the leading termof $H$ times $ dQ sub ! j open . $ parenthesis\quad Inthecase f sub 1 to the power $m of lambda = 0 sub 1 ,$ period i.e.,for period period f sub j to the power of lambda subthe j plus Weyl 1 period algebra period $period A { f ton the} power( Kof lambda ) $ sub one q closing has the parenthesis fundamental = b open theorem parenthesis of lambda J . closing Bernstein that parenthesisasserts f that sub 1 to , thefor power any of non lambda− t r sub i v i 1 a periodl $ A period{ n period} (K) f to the power of− lambda$ module sub q open $M$ parenthesis so that j gr =$( 1 comma M period )$ period is of period finite comma type q closing , parenthesis hold period \ [ d ( M ) \geq n . \ ]

\noindent An $ A { n } (K) − $ module $M$ suchthat $d ( M ) = n $ i s said to b e holonomic .

One of the applications of the concept of holonomic modules i s the existence of the Bernstein −− Sato functional equations [ 1 7 , \quad 25 , \quad 23 ] , \quad i . e . , \quad given polynomials $ f 1 , . . . , f { q }$ in $ K [ x { 1 } , . . . , x { n } ] $ \quad there are differential operators $ Q { j }$ \quad in $ A { n } (K [ \lambda ] ) , $ with

\noindent $ \lambda = ( \lambda { 1 } ,..., \lambda { q } ) , $ and a non − zero polynomial $ b \ in K[ \lambda ] $ such that the formal rela − t i o n s

\ [Q { j } ( f ˆ{\lambda { 1 }} { 1 } . . . f ˆ{\lambda { j } + 1 } { j } . . . f ˆ{\lambda { q }} ) = b ( \lambda ) f ˆ{\lambda { 1 }} { 1 } . . . f ˆ{\lambda { q }} (j=1,...,q) \ ]

\noindent hold . 82 .. C period A period BERENSTEIN AND A period YGER \noindentWe are interested82 \quad in theC following . A . BERENSTEIN E sub n comma AND m open A . parenthesis YGER K closing parenthesis hyphen modules comma with m less or equal n period .. Consider ex hyphen Weponential are interested polynomials in P the sub 1 following comma period $ period E { periodn , comma m } P sub(K) q of n variables− $ with modules positive , integral with frequencies$ m \ leq n . $ \quad Consider ex − 82 C . A . BERENSTEIN AND A . YGER ponentialand coefficients polynomials in a subfield K $ of P C{ comma1 } that,...,P i s comma finite sums { q }$ of $ n $ variables with positive integral frequencies We are interested in the following E ( )− modules , with m ≤ n. Consider ex - po- andLine coefficients 1 P sub j open parenthesis in a subfield x closing $K$parenthesis ofn,m =K sum $C c sub , j $ comma that k open i s parenthesis , finite x sums closing parenthesis nential polynomials P , ..., P of n variables with positive integral frequencies and coefficients e to the power of k times x comma Line1 2 kq in N m in a subfield of , that i s , finite sums \ [ \withbegin c sub{ a l ji g comma n e d } kP inK{ K openj C} square( x bracket ) x =closing\sum square bracketc { j comma , j = k 1} comma( period x ) period e ˆperiod{ k \cdot x } , \\ comma q period .. We consider a new field K open parenthesisX lambda closingk·x parenthesis = K open parenthesis lambda Pj(x) = cj,k(x)e , subk 1 comma\ in periodN period m \end period{ a l i gcomma n e d }\ lambda] sub q closing parenthesis obtained from K by adjoining q indeterminates comma and definek ∈ theNm module M freely generated by a single generator denoted P to the power of lambda = P sub 1 to the power of lambda sub 1 period with cj,k ∈ K[x], j = 1, ..., q. We consider a new field K(λ) = K(λ1, ..., λq) period\noindent periodwith P sub q $ to c the{ powerj of , lambda k }\ subin q commaK[x],j=1,... namely comma , q . $ obtained\quad fromWeK considerby adjoining a newq indeterminates field $K , and ( define\lambda the module M)freely = K ( \lambda { 1 } Equation: open parenthesis 3 closing parenthesis .. M = M openλ parenthesis P sub 1 comma period period period generated by a single generator denoted Pλ = Pλ1 ...P q , namely , comma,..., P sub q closing parenthesis\lambda : = K{ openq } parenthesis) $ lambda1 closingq parenthesis open square bracket x sub 1 comma periodobtained period from period comma $ K $ line-x by sub adjoining n comma e $ to q the $ power indeterminates of x sub 1 comma , period and define period period the modulecomma line-e $M$ to f r e e l y the power of x sub m line-bracketright open square bracket 1x1 slash P subx 1m comma period period period commaλ 1 slash M = M(P1, ..., Pq) := K(λ)[x1, ..., line − xn,e , ..., line − e line − bracketright[1/P1, ..., 1/Pq]P , P sub q closing square bracket P to the power of lambda comma (3) \noindentwhere commagenerated t o pick up by the aearlier single notation generator comma denoted X sub i open $ P parenthesis ˆ{\lambda resp period} = comma P ˆ{\ Y sublambda j closing{ 1 }} { 1 } . . . P ˆ{\lambda { q }} { q } , $ namely , parenthesis operateswhere as , tmultiplication o pick up the earlier notation ,Xi ( resp ., Yj) operates as multiplication by xi( xj by x sub i openresp parenthesis . , by e ) and respD periodj acts ascomma the differential by e to the operator power of∇j x, subdefined j closing by parenthesis and D sub j acts as the\ begin differential{ a l i g n operator∗} nabla sub j comma defined by \ tagnabla∗{$ sub ( j line-parenleft 3 ) $} M A P to = the M power ( of lambda P { closing1 } ,...,P parenthesisk=1 : = parenleftbigg sub{ q partialdiff} ) sub : x sub = j K( \lambda ) [ x { 1 } , .λ .∂A . ,X lk i n eP−k x {λ n , } e ˆ{ x { 1 }} to the power of partialdiff A plus∇ Aj sumline − fromparenleftA k = 1 toP q lambda) := (∂ Pxj sub+ A k toλ theP k∂ power∂ xj )P of. k partialdiff sub partialdiff sub x, sub . j to the . power . of , P sub l i n k e parenrightbigg−e ˆ{ x { Pm to}} the powerl i n e − ofbracketright lambdaq period [ 1 / P { 1 } ,. . . , 1 / P { q } ] P ˆ{\lambda } , The naturalThe filtration natural of filtrationM is of M is \endCapital{ a l i g Gamma n ∗} sub v : = braceleftbigg R open parenthesis sub P sub 1 to the power of open parenthesis lambda sub (λ , , ex) λ x period to the power of comma xΓ period:= {R period( x sub) Pv P to the: R power∈ (λ)[ ofx, comma e ], deg subx qR to≤ thevd } power, of closing parenthesis to v P1 . ..P q K x,e 0 the\noindent power of vwhere to the power , t o of picke to the up power the of earlier x closing parenthesis notation P to $ the , power X { ofi lambda} ( : $ R in resp K open $ parenthesis . , Ylambda{ j closing} )where parenthesis $ operatesd0 := open 1 + deg square asx,e multiplicationx (P bracket1...Pq). x commaThis i se ato good the power filtration of x and closing square bracket comma degree sub x by $ x { i } ($ resp. ,by $eˆ{ x { j }} ) $ and $ D { j }$ acts as the differential operator comma e to the power of x R less or equal vd sub 0 bracerightbigg comma  $ \nabla { j } , $ defined by n + m + vd0 where d sub 0 : = 1 plus degree sub x commadim ( eλ) toΓv the= power of x open parenthesis. P sub 1 period period period P K vd0 sub q closing parenthesis period .. This i s a good filtration and \ [ dimension\nabla sub{Hencej K} open , l i parenthesis n e −parenleft lambda closing A Pˆ parenthesis{\lambda Capital} Gamma) : sub = v = Row( {\ 1 npartial plus m plus}ˆ vd{\ subpartial A0 Row} { 2x vd sub{ j 0}} . period+ A \sum ˆ{ k = 1 } { q }\lambda { P }ˆ{ k } { k }\ partial {\ partial }ˆ{ P { k }} { x { j }} ) P ˆ{\lambda } . \ ] n+m Hence comma d(M) = n + m, e(M) = d0 . d open parenthesis M closing parenthesis = n plus m comma e open parenthesis M closing parenthesis = d sub 0 to It is natural t o ask whether for every non - trivial En,m(K)− the power of n plusmodule m period ( or \noindentIt .. is .. naturalThe natural .. t o .. ask filtration .. whether .. of for .. $M$ every .. nonis hyphen trivial .. E sub n comma m open parenthesis K En,m(K(λ))− module ) with m ≤ n, one has d(M) ≥ n+m. Or , at least , t o give conditions closing parenthesisthat hyphen ensure modulethis inequality .. open occurs parenthesis . One or can give examples showing that this may depend on \ [ E\Gamma sub n comma{ v m} open: parenthesis = \{ KR open{ ( parenthesis}ˆ{ ( lambda\lambda closing} { parenthesisP { 1 }} closingˆ{ , parenthesis} { . } hyphenx { module.. }ˆ{ , } { P }ˆ{ e ˆ{ x } the choice of field K. closing) } { parenthesisq ˆ{ ) ˆ{ withv m}}} lessP or equalˆ{\lambda n comma} one:R has d open\ in parenthesisK( M closing\lambda parenthesis) greater [ equalx ,n plus e ˆ{ x } ], \deg In{ thex case ,m = e 1 ˆ,{asx a substitute}} R for\ leq Bernsteinvd ’ s{ theorem0 }\} we have, the\ ] follow - ing result m period .. Or. comma at least comma t o give conditions that ensure this inequality occurs period One can give examples showing that Proposition 1 . 1 . Let M be a finitely generated En,1(K)− module , then , either this may depend on the choice of field K period d(M) ≥ n + 1 or for every e lement m0 ∈ M \{0} there exist two \noindentIn the casewhere m = 1 comma $ d as{ a0 substitute} : = for Bernstein 1 + quoteright\deg { s theoremx , we e have ˆ{ thex follow}} (P hyphen { 1 } . non - zero polynomials A, B ∈ K[s], and t ∈ N such that ..Ping result period{ q } ) . $ \quad This i s a good filtration and Proposition 1 period 1 period Let M be a finitelyt generated E sub n comma 1 open parenthesis K closing parenthesis Y A(X1)m0 = B(Y1)m0 = 0. hyphen\ [ \dim module{ K( comma then\lambda comma either) }\Gamma1 { v } = \ l e f t (\ begin { array }{ c} n + m + vd d{ open0 parenthesis}\\ vd M{ closing0 }\end parenthesisAs{ array an application}\ greaterright ofequal). Proposition n\ plus] 1 1.. . or 1 for t o .. the every module .. e lement m sub 0 in M backslash open brace 0 closing brace .. there .. exist .. two .. non hyphen zero x1 λ polynomials A comma B in K openM(P square1, ..., Pq bracket) = K(λ s)[ closingx1, ..., x squaren, e ][1 bracket/P1, ..., comma1/Pq]P .. and t in N such that \noindent Hence , Y sub 1 to the power of t A open parenthesis X sub 1 closingx parenthesis1 m sub 0 = B open parenthesis Y sub 1 defined by equation ( 2 ) , where Pj ∈ K[x1, ..., xn, e ], K a subfield of C, one can prove the closing parenthesisfollowing m sub proposition 0 = 0 period . \ [d(M)=n+m,e(M)=dˆAs an application of Proposition 1 period 1 t o the module { n + m } { 0 } . \ ] M open parenthesis P sub 1 comma period period period comma P sub q closing parenthesis = K open parenthesis lambda closing parenthesis open square bracket x sub 1 comma period period period comma x sub n comma e to the power\ hspace of x∗{\ subf1 i l closing l } I t square\quad bracketi s \quad open squarenatural bracket\quad 1 slasht o P\ subquad 1 commaask \ periodquad periodwhether period\quad commaf o 1 r slash\quad every \quad non − t r i v i a l \quad P$ sub E q{ closingn , square m bracket} (K) P to the power− $ of lambda module \quad ( or defined by equation open parenthesis 2 closing parenthesis comma where P sub j in K open square bracket x sub 1 comma\noindent period period$ E { periodn comma , m x} sub(K( n comma e to the\lambda power of x sub)) 1 closing− square$ module bracket ) comma with K a$m subfield\ leq ofn C comma ,$ onehas one can $d ( M ) \geq n + m . $ \quad Or , at least , t o give conditionsprove the following that proposition ensure this period inequality occurs . One can give examples showing that this may depend on the choice of field $K . $

In the case $m = 1 , $ as a substitute for Bernstein ’ s theorem we have the follow − ing r e s u l t .

\ hspace ∗{\ f i l l } Proposition 1 . 1 . Let $M$ be a finitely generated $ E { n , 1 } (K) − $ module , then , either

\noindent $ d ( M ) \geq n + 1 $ \quad or f o r \quad every \quad e lement $ m { 0 }\ in M \setminus \{ 0 \} $ \quad ther e \quad e x i s t \quad two \quad non − zero polynomials $A , B \ in K [ s ] , $ \quad and $ t \ in N $ such that

\ [ Y ˆ{ t } { 1 } A(X { 1 } ) m { 0 } = B ( Y { 1 } ) m { 0 } = 0 . \ ]

\ centerline {As an application of Proposition 1 . 1 t o the module }

\ [M(P { 1 } ,...,P { q } ) = K ( \lambda ) [ x { 1 } , . . . , x { n } , e ˆ{ x { 1 }} ] [ 1 / P { 1 } ,.. . , 1 / P { q } ] P ˆ{\lambda }\ ]

\noindent defined by equation ( 2 ) , where $ P { j }\ in K [ x { 1 } ,. . . , x { n } , e ˆ{ x { 1 }} ] , K$ asubfield of $C ,$ onecan prove the following proposition . EXPONENTIAL POLYNOMIALS .. 83 \ hspaceProposition∗{\ f i 1 lperiod l }EXPONENTIAL 2 period There POLYNOMIALS are two non\ hyphenquad 83 zero polynomials A sub 1 comma A sub 2 of a single vari hyphen Propositionable s comma 1 with . 2 coefficients . There are in K two open non square− zero bracket polynomials lambda closing $ square A { bracket1 } comma,A lambda{ 2 }$ = open of a single vari − parenthesisable $ lambda s , sub $ 1 with comma coefficients period period period in $K comma [ lambda\lambda sub q closing], parenthesis\lambda comma and= 2 (q linear\lambda { 1 } EXPONENTIAL POLYNOMIALS 83 differential,..., opera hyphen \lambda { q } ) , $ and $ 2 q $ linear differential opera − Proposition 1 . 2 . There are two non - zero polynomials A ,A of a single vari - able tors comma Q i comma j open parenthesis i = 1 comma 2 semicolon j = 11 comma2 period period period comma q s, with coefficients in [λ], λ = (λ , ..., λ ), and 2q linear differential opera - closing\noindent parenthesistors$,Qi comma with coefficientsK , belonging j1 ( toq K i=1 open square bracket ,2; lambda comma j=1 x comma e , to the . power . tors , Qi, j(i = 1, 2; j = 1, ..., q), with coefficients belonging to [λ, x, ex1 , e−x1 ], such that of. x sub , 1 comma q ) e to ,$ the power with of coefficients minus x sub 1 closing belonging square bracket to $K commaK [ \lambda , x , e ˆ{ x { 1 }} for every j, ,such e ˆ that{ − for everyx { j comma1 }} ] , $ suchEquation: that open for parenthesis every $ 4 closing j , parenthesis $ .. A sub 1 open parenthesis lambda comma x sub 1 closing parenthesis λ (λ x1 −x1 λ P to the power of lambda = Q 1 commaA j1 to(λ, the x1) powerP = Q of1 open, j, x, parenthesis e , e , ∂/∂x lambda)PjP sub, comma x comma e(4) to the power of x\ begin sub 1 comma{ a l i g n e∗} to the power of minus x sub 1 comma partialdiff slash partialdiff x closing parenthesis P sub j P to the A (λ, ex1 )Pλ = Q2, j(λx, ex1 , e−x1 , ∂/∂x)P Pλ. (5) power\ tag ∗{ of$ lambda ( 4 comma ) $ Equation:} A { 1 open} 2 parenthesis( \lambda 5 closing, , parenthesis x { 1 ..} A subj) 2 open P ˆ{\ parenthesislambda lambda} = comma Q 1e to, the j power ˆ{ ( of x\ sublambda 1 closing} { parenthesis, } x P to , the e power ˆ{ x of lambda{ 1 }} = Q, 2 comma e ˆ{ j− to thex power{ 1 of}} open, parenthesis\ partial / \ partial( To simplifyx ) the P notation{ j } weP have ˆ{\ writtenlambda∂/∂x} to, denote\\\ tag(∂/∂x∗{$1, ( ..., ∂/∂x 5n). )) $} A { 2 } ( lambda sub comma2. x commaF − u nctional e to the power equations of x sub . 1 commaAs explained e to the in[ power 3 ] , t of o studyminus B xe ´subzout 1 comma identities partialdiff , slash partialdiff\lambda x closing, e parenthesis ˆ{ x { P1 sub}} j P) to the P power ˆ{\lambda of lambda} period= Q 2 , j ˆ{ ( \lambda } { , } x , e ˆ{divisionx { problems1 }} , , and e ˆ the{ − like ,x one{ 1 needs}} , to determine\ partial the principal/ \ partial part of x ) P { j } open parenthesis To simplify the notation we2 haveλ written partialdiff slash partialdiff x to denote open parenthesis par- the Laurent development of | f | for λ = −k, k ∈ N, where f is an exponen - tial Ptialdiff ˆ{\ slashlambda partialdiff} . x sub 1 comma period period period comma partialdiff slash partialdiff x sub n closing parenthesis \end{ a l i g n ∗}polynomial . The reason for this need will become clear lat er on . The fol - period closing parenthesislowing lemma , a consequence of Proposition 1 . 2 , provides some of this informa - 2 period F-ution nctional . equations period .. As explained in open square bracket 3 closing square bracket comma t o study\noindent B e-acute( Tozout simplify identities comma the notation we have written $ \ partial / \ partial x $ to denote Lemma 2 . 1 . Let f be an exponential polynomial in En,1(K), k ∈ N, there is an $ (division\ partial problems comma/ \ ..partial and the likex comma{ 1 ..} one,..., needs .. to .. determine the\ partial principal part/ .. of\ partial x { n } )integer .q )∈ $N such that for any N ∈ N one can find a non - zero polynomial RN ∈ K[x1] the Laurentand development a functional of bar equation f bar to of the the power form of 2 lambda for lambda = minus k comma k in N comma .. where f is an exponen hyphen $tial 2 polynomial . F−u period $ nctional .. The reason equations for this need . \ willquad becomeAsexplained clear lat er on in period [ 3 .. ] The , t fol o hyphen study B $ \acute{e} $ zout identities , q 2λ 2λ k+1 q+N 2λ lowing lemma comma a consequence(λ + k) RN of| f Proposition| = Qk, N 1(| periodf | f 2 comma) + (λ provides+ k) somevN | f of| this, informa hyphen(6) divisiontion period problems , \quad and the like , \quad one needs \quad to \quad determine the principal part \quad o f Lemma 2 periodwhere 1v periodN ∈ K[ Letλ, x1 f] .. be anand exponentialQk, N is polynomial a linear differential in E sub operator n comma with 1 open coefficients parenthesis in K closing parenthesis\noindent commathe Laurent k in N comma development .. there is of $ \mid f \mid ˆ{ 2 \lambda }$ f o r $ \lambda = − k , k \ in N , $ \quad where $ f $ is an exponen − an integer q in N such that for any N in N .. one can findx a non−x hyphen zero polynomial K[λ, x, e 1 , e 1 ]. tialR sub polynomial N in K open square . \quad bracketThe x reason sub 1 closing for square this bracket need will .. and become a functional clear equation lat of er the on form . \quad The f o l − Equation: open parenthesis 6 closingMore generally parenthesis , one .. open can obtain parenthesis relations lambda of the plus form k closing parenthesis to the power of\noindent q R sub N barlowing f bar lemma to the power , a ofconsequence 2 lambda = Q of k comma Proposition N open parenthesis 1 . 2 , barprovides f bar to some the power ofof this 2 lambda informa − f to the power of k plus 1 closing parenthesis plus open parenthesis lambda plus k closing parenthesis to the power of q q˜ x1 2λ ˜ 2λ 2k+1 q˜+N 2λ plus\noindent N v sub Nt ibar o n f . bar to(λ the+ k power) SN ( ofe 2) lambda| f | = commaQk, N(| f | f ) + (λ + k) v˜N | f | , (7) where v sub N in K open square bracket lambda comma x sub 1 closing square bracket .... and Q k comma N is a x1 ˜ Lemma2linear differential . 1where . operator Letv ˜N ∈ $ withK f[λ, $ coefficients e \],SquadN (t) inbe = S anN,k(t exponential) ∈ K[t], and Qk, polynomial N is a differential in $ operator E { n with, the 1 } ( KK ) open ,squaresame k bracket properties\ in lambdaN as Qk, comma , $ N. \ xquad commathe e reto the i s power of x sub 1 comma e to the power of minus x sub 1 an integer $ q The\ in way oneN$ uses such these relationsthat for i s anythe following $N .\ in One knowsN $ a\ prioriquad [one 1 ] that can , find a non − zero polynomial closing square bracket period 2λ $More R generally{ N }\in a comma neighborhoodin oneK can [ of obtainλ x= − relations{k,1the} distribution of] the $ form\quad - valuedand meromorphic a functional function equation| f | has of the the form Equation: openLaurent parenthesis expansion 7 closing parenthesis .. open parenthesis lambda plus k closing parenthesis to the power of\ begin q-tilde{ Sa l sub i g n N∗} open parenthesis e to the power of x sub 1 closing parenthesis bar f bar to the power of 2 lambda = \ tag ∗{$ ( 6 ) $} ( \lambda + k ) ˆ{ q } R { N }\mid f \mid ˆ{ 2 \lambda } tilde-Q k comma N open parenthesis bar f bar to the power of 2 lambda f∞ to the power of 2 k plus 1 closing parenthesis =plus open Q parenthesis k , Nlambda ( plus\mid k closingf parenthesis\mid ˆ to{ the2 power\lambda of q-tilde} plusf ˆ N{ tilde-vk + sub N 1 bar} f bar) to + the power ( \lambda + k ) ˆ{ q + N } v 2λ { NX}\mid f j \mid ˆ{ 2 \lambda } , of 2 lambda comma | f | = ak,j(λ + k) , (8) \end{ a l i g n ∗} where v-tilde sub N in K open square bracket lambda comma e toj = the− power2n of x sub 1 closing square bracket comma S sub N open parenthesis t closing parenthesis = S sub N comma k open parenthesis t closing parenthesis in K open \noindent where $ v { N0 }\n in K[ \lambda , x { 1 } ] $ \ h f i l l and $ Q square bracket twith closingak,j square∈ bracketD (C ). commaThe previous and tilde-Q lemma k comma allows N us is to a differentialcompute explicitly operator the products k , N $ is a linearx1 differential operator with coefficients in with the sameRN properties(x1)ak,j,S asN (e Q k)a commak,j, for N period−2n ≤ j ≤ 0, if we let N = 2n + 1. The way oneNamely uses these , the relations polynomial i s thevN followingin the statement period .. of One Lemma knows 2 . a 1 priori can b open e expanded square inbracket 1 closing square bracket\ begin that{ a l i comma g npowers∗} of λ + k, i . e . , K[in a neighborhood\lambda of lambda, x = minus , k e comma ˆ{ x the{ distribution1 }} , hyphen e ˆ{ valued − x meromorphic{ 1 }} function]. bar f bar to the power\end{ ofa l 2 i g lambda n ∗} has the Laurent expansion m \ centerline {More generally , one can obtainX relations ofl the form } infinity Equation: open parenthesis 8 closingvN ( parenthesisλ, x1) = ..vN,l bar(x f1 bar)(λ + tok the) . power of 2 lambda = sum(9) a sub k comma j open parenthesis lambda plus k closing parenthesis to the power of j comma j = minus 2 n \ beginwith{ aa sub l i g nk∗} comma j in D to the power of prime open parenthesis Cl to= 0the power of n closing parenthesis period .. The\ tag previous∗{$ ( lemma 7 allows ) $} us( to compute\lambda explicitly+ the k ) ˆ{\ tilde {q}} S { N } ( e ˆ{ x { 1 }} ) products\mid .. Rf sub\mid N openˆ{ parenthesis2 \lambda x sub} 1 closing= parenthesis\ tilde {Q} a subk k comma , N j comma ( S\ submid N openf parenthesis\mid ˆ{ 2 e\lambda to the power} f of ˆx{ sub2 1 closing k + parenthesis 1 } ) a sub + k comma ( \ jlambda comma .. for+ .. minus k ) 2 nˆ{\ lesstilde or equal{q} j less+ or equal N } 0 comma\ tilde ..{v if} we{ letN ..}\ N =mid 2 n plusf 1 period\mid ˆ{ 2 \lambda } , \endNamely{ a l i g comma n ∗} the polynomial v sub N in the statement of Lemma 2 period 1 can b e expanded in powers of lambda plus k comma i period e period comma \noindentm Equation:where open $parenthesis\ tilde { 9v} closing{ N parenthesis}\ in ..K[ v sub N\ openlambda parenthesis, lambda e ˆ{ x comma{ 1 x}} sub], 1 closing parenthesisS { N } =( sum v t sub ) N comma = lS open{ N parenthesis , k x sub} 1( closing t parenthesis ) \ in openK parenthesis [ t lambda ] plus , $ k closingand parenthesis$ \ tilde { toQ} the powerk ,of l period N$ l is = 0 a differential operator with the same properties as $Q k , N . $

\ hspace ∗{\ f i l l }The way one uses these relations i s the following . \quad One knows a priori [ 1 ] that ,

\noindent in a neighborhood of $ \lambda = − k , $ the distribution − valued meromorphic function $ \mid f \mid ˆ{ 2 \lambda }$ has the Laurent expansion

\ begin { a l i g n ∗} \ infty \\\ tag ∗{$ ( 8 ) $}\mid f \mid ˆ{ 2 \lambda } = \sum a { k , j } ( \lambda + k ) ˆ{ j } , \\ j = − 2 n \end{ a l i g n ∗}

\noindent with $ a { k , j }\ in D ˆ{\prime } ( C ˆ{ n } ) . $ \quad The previous lemma allows us to compute explicitly the products \quad $ R { N } ( x { 1 } ) a { k , j } ,S { N } ( e ˆ{ x { 1 }} ) a { k , j } , $ \quad f o r \quad $ − 2 n \ leq j \ leq 0 , $ \quad i f we l e t \quad $ N = 2 n + 1 . $ Namely , the polynomial $ v { N }$ in the statement of Lemma 2 . 1 can b e expanded in

\noindent powers o f $ \lambda + k , $ i . e . ,

\ begin { a l i g n ∗} m \\\ tag ∗{$ ( 9 ) $} v { N } ( \lambda , x { 1 } ) = \sum v { N , l } ( x { 1 } )( \lambda + k ) ˆ{ l } . \\ l = 0 \end{ a l i g n ∗} 84 .. C period A period BERENSTEIN AND A period YGER \noindentLet phi in84 D to\quad the powerC . ofA prime . BERENSTEIN open parenthesis AND A C . to YGER the power of n closing parenthesis comma then infinity Equation: open parenthesis 1 0 closing parenthesis .. open parenthesis lambda plus k closing parenthesis to the power of\ begin q angbracketleft{ a l i g n ∗} bar f bar to the power of 2 lambda comma R sub N open parenthesis x sub 1 closing parenthesis phi rightLet angbracket\phi =\ sumin angbracketleftD ˆ{\prime a sub} k comma( C j ˆ comma{ n } R) sub N , phi rightthen angbracket\\\ infty open\\\ parenthesistag ∗{$ ( lambda 1 84 C . A . BERENSTEIN AND A . YGER plus0 k ) closing $} ( parenthesis\lambda to the+ power k of q ) plus ˆ{ jq j =}\ minuslangle 2 n = angbracketleft\mid f Q k\ commamid ˆ{ N open2 parenthesis\lambda } lambda, Rclosing{ N parenthesis} ( x open{ parenthesis1 } ) bar\phi f bar to\rangle the power of= 2 lambda\sum f to\ thelangle power ofa 2 k{ plusk 1 closing , j parenthesis} , Rcomma{ N phi}\ rightphi angbracket\rangle infinity plus( sum\lambda angbracketleft+ a k sub k ) comma ˆLet{ qφ ∈ j commaD +0( n) j, vthen}\\ sub Nj phi =right− angbracket2 n open\\ = \ langle Q k , N ( \lambda )( \mid f \Cmid ˆ{ 2 \lambda } f ˆ{ 2 parenthesis lambda plus k closing parenthesis to the power of q plus N plus j j = minus∞ 2 n = angbracketleft bar f bar tok the + power 1 of} 2 lambda), f to\phi the power\rangle of 2 k plus\\\ 1 commainfty Q\\ sub+ k comma\sum N to\ langle the power ofa prime{ k open , parenthesis j } , v { N }\phi \rangle q ( 2λ \lambda +X k ) ˆ{ q +q+j N + j }\\ j = lambda closing parenthesis phi right(λ + angbracketk) h| f | ,R plusN (x sum1)φi angbracketleft= hak,j a,R subN φi k(λ comma+ k) j comma v sub(10) N comma l phi −right angbracket2 n \\ open= parenthesis\ langle lambda\mid plus kf closing\mid parenthesisˆ{ 2 to\lambda the power} of qf plus ˆ{ N2 plus k j plus + l comma 1 } j comma, Q ˆ{\prime } { k ,N } ( \lambda ) \phi \rangle \\ + \sum \ langlej = −2na { k , j } , l 2λ 2k+1 v where{ N Q sub, k l comma}\phi N to the\rangle power of prime( i s\lambda the adj= ointhQk,+ operator N(λ k)(| f of| ) Q ˆf k{ commaq), φi + N open N parenthesis + j obtained + l } by, \\ integrationj , byparts l closing parenthesis period ∞ \endThe{ a first l i g n t∗} erm of the last sum i s holomorphic at lambdaX = minus k comma and the series only + ha , v φi(λ + k)q+N+j contains powers of lambda plus k bigger than or equal t o q plus 1k,j commaN due t o the choice of N period \noindentThus commawhere the distribution $ Q ˆ{\ hyphenprime valued} { functionk , open N } parenthesis$ i s the lambda adj plusj oint= − k2 closingn operator parenthesis of to $Q the power k , N ( $ obtained by integration by parts ) . of q R sub N open parenthesis x sub 1 closing parenthesis bar f bar to the2λ power2k+1 of0 2 lambda is holomorphic in a = h| f | f ,Qk,N (λ)φi neighboorhood of lambda = minus k period .. Moreover comma if we denote The first t erm of the last sum i s holomorphicX at $ \lambdaq+N+j=+l − k , $ and the series only infinity Equation: open parenthesis 1 1 closing parenthesis+ h ..ak,j open, vN,l parenthesisφi(λ + k) lambda, plus k closing parenthesis to contains powers of $ \lambda + k$ bigger than or equal to $q + 1 ,$ duetothe choice of the power of q R sub N open parenthesis x sub 1 closing parenthesis bar f bar to the powerj, l of 2 lambda = sum b sub k comma$ N h . open $ parenthesis lambda plus k closing parenthesis to the power of h h = 0 0 it s Taylor developmentwhere Qk,N i comma s the adj then oint comma operator for of 0Qk, less N or( equalobtained h less by integrationor equal q comma by parts the ) . distributions b sub k comma\noindent h are givenThusThe by , thefirst t distribution erm of the last sum− valued i s holomorphic function at λ = $− ( k, and\lambda the series+ only contains k ) ˆ{ q } R { N } (Equation: x { 1 open}powers) parenthesis of\λmid+ k 1bigger 2f closing than\mid parenthesis or equalˆ{ 2 t o ..\q angbracketleftlambda+ 1, due t}$ o the b is sub choice holomorphic k comma of N. h comma in aphi right angbracket = neighboorhood of $ \lambda = − k . $ q \quad Moreover2λ , if we denote 2 1 pi i bar integralThus k , plus the lambda distribution bar = - valuedepsilon function integral C (λ to+ thek) R powerN (x1) of| nf bar| fis open holomorphic parenthesis in xaclosing neigh- parenthesis bar to the powerboorhood of 2 lambda of λ f= open−k. parenthesisMoreover x , closing if we denote parenthesis to the power of 2 k plus 1 Q sub k comma N to the power\ begin of{ primea l i g n open∗} parenthesis lambda closing parenthesis open parenthesis phi open parenthesis x closing parenthesis closing\ infty parenthesis\\\ tag ∗{ dx$ open ( parenthesis 1 1 lambda ) $} d( plus\lambda k sub closing+ parenthesis k ) to ˆ{ theq power} R of{ h plusN } 1 to( the x power{ 1 of } ∞ lambda) \mid commaf \mid ˆ{ 2 \lambda } = \sum b { k , h } ( \lambda + k ) ˆ{ h }\\ h = 0 q 2λ X h where epsilon greater 0 is chosen sufficiently(λ + k small) RN ( sx o-line1) | f that| = on ab neighborhoodk,h(λ + k) of supp open parenthesis(11) phi closing parenthesis\end{ a l i g comman ∗} h = 0 the line-f sub unction x mapsto-arrowright bar f open parenthesis x closing parenthesis bar to the power of minus 2 \noindent it s Taylor development , then , for $ 0 \ leq h \ leq q , $ the distributions epsilon i s integrableit s Taylor period development , then , for 0 ≤ h ≤ q, the distributions bk,h are given by $ bWe{ cank rewrite , the h last}$ integral are given as by Line 1 2 1 pi i bar integral plus lambdaZ k barZ = epsilon integral C to the power of n line-bar f line-parenleft x \ begin { a l i g n ∗} 2λ 2k+1 0 λ closing parenthesis bar tohbk,h the, φ poweri = 21 ofπi 2| open parenthesisε | f(x) lambda| f(x plus) kQ closingk,N (λ)( parenthesisφ(x))dx(λd+ fk open)h+1 , parenthesis(12) x closing \ tag ∗{$ ( 1 2 ) $}\ langlek |= b n{ k , h } , \phi \rangle = 2 1 {\ pi } parenthesis open parenthesis f open parenthesis+λ x closingC parenthesis slash f open parenthesis x closing parenthesis closing parenthesisi \mid to the\ int power{ k of k{ Q+ sub{\ k commalambda N}}\ to themid power= of prime}\ varepsilon open parenthesis\ lambdaint { closingC ˆ{ parenthesisn }}\mid f ( xwhere ) \εmid > 0 isˆ{ chosen2 sufficiently\lambda small} f s o − (line x that on ) ˆ a{ neighborhood2 k + of supp 1 } (φ)Q, ˆ{\prime } { k open parenthesis phi open parenthesis x closing−2ε parenthesis closing parenthesis dx open parenthesis lambda d plus k sub ,N } ( the\lambdaline − function)(x 7→| f(x)\|phi i s integrable( x . ) ) dx ( \lambda d { + k }ˆ{\lambda } { ) ˆ{ h closing parenthesis to the power of h plus 1 toWe the can power rewrite of lambdathe last Lineintegral 2 = as sum from j = 0 to infinity 1 j ! 2 1 pi i bar +integral 1 plus}} lambda, k bar = epsilon integral C to the power of n open parenthesis log bar f bar to the power of 2 closing \end{ a l i g n ∗} parenthesis to the powerZ of j fZ open parenthesis f slash f closing parenthesis to the power of k Q sub k comma N to the 2(λ+k) k 0 λ power of prime open21πi | parenthesisε lambdaline − closingbarfline parenthesis− parenleftx open) | parenthesisf(x)(f phi(x) open/f(x)) parenthesisQk,N (λ)(φ x(x closing))dx(λd parenthesis+k)h+1 \noindent where $ \ varepsilonn > 0 $ is chosen sufficiently small s $ o−line $ that on a neighborhood of supp closing parenthesis dx open+λk|= parenthesisC lambda plus k closing parenthesis to the power of j minus h minus 1 d lambda $ ( \phi ) , $ comma j=0 Z Z X 2 j k 0 j−h−1 which shows that the t erms .. angbracketleft= 1j!21π bi | sub k commaε (log h comma| f | ) phif(f/f right) angbracketQk,N (λ)(φ( ..x)) aredx( linearλ + k) combinationsdλ, \noindent the $ l i n e −f { unction } x+ k|=\mapston \mid f ( x ) \mid ˆ{ − 2 of integrals of the ∞ λ C \ varepsilonform }$ i s integrable . which shows that the t erms hbk,h, φi are linear combinations of integrals of the form Equation: open parenthesis 1 3 closing parenthesis .. integral open parenthesis log bar f bar to the power of 2 closing \ centerline {We can rewrite the last integral as } parenthesis to the power of j f open parenthesisZ f slash f closing parenthesis to the power of k Q sub iota open parenthesis 2 j k phi closing parenthesis dx comma C to the power(log of n| f | ) f(f/f) Qι(φ)dx, (13) \ [ \wherebegin j{ ina Nl i g open n e d parenthesis} 2 1 in{\ factpi comma} i 0 less\mid or equal\ int j less or{ equal+ {\ h pluslambda 1 closing} parenthesisk \mid and= the}\ Q subvarepsilon n iota\ int are{ differentialC ˆ{ n operators}} l i n ewith−bar f l i n e −parenleft xC ) \mid ˆ{ 2 ( \lambda + kcoefficients ) } f(x)(f(x)/f(x))ˆ in K open square bracket x comma e to the power of x sub 1 comma e to the power{ k of} minusQ ˆ x{\ subprime 1 closing} { k where j ∈ ( in fact , 0 ≤ j ≤ h + 1) and the Q are differential operators with coefficients in square,N bracket} ( period\lambda NoteN that)( the term\ openphi parenthesis( x fι slash ) f ) closing dx parenthesis ( \lambda to the powerd of{ k+ i s bounded k }ˆ{\lambda } { ) ˆ{ h [x, ex1 , e−x1 ]. Note that the term (f/f)k i s bounded , and the same holds locally for f( log +comma 1 and}}\\ theK same | f |2)j. =holds\ locallysum ˆ{ forj f open = parenthesis 0 } {\ loginfty bar f} bar1{ to thej } power! of 2 2 closing 1 {\ parenthesispi } i to the\mid power of\ jint period{ + {\lambda } k \mid = }\ varepsilon \ int { C ˆ{ n }} ( \ log \mid f \mid ˆ{ 2 } ) ˆ{ j } f ( f / f ) ˆ{ k } Q ˆ{\prime } { k , N } ( \lambda )( \phi ( x ) ) dx ( \lambda + k ) ˆ{ j − h − 1 } d \lambda , \end{ a l i g n e d }\ ]

\noindent which shows that the t erms \quad $ \ langle b { k , h } , \phi \rangle $ \quad are linear combinations of integrals of the form

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 3 ) $}\ int ( \ log \mid f \mid ˆ{ 2 } ) ˆ{ j } f ( f / f ) ˆ{ k } Q {\ iota } ( \phi ) dx , \\ C ˆ{ n } \end{ a l i g n ∗}

\noindent where $ j \ in N ($ infact $, 0 \ leq j \ leq h + 1 ) $ and the $ Q {\ iota }$ are differential operators with coefficientsin $K [ x , eˆ{ x { 1 }} , e ˆ{ − x { 1 }} ] . $ Note that the term $ ( f / f ) ˆ{ k }$ i s bounded , and the same holds locally for $ f ( $ log $ \mid f \mid ˆ{ 2 } ) ˆ{ j } . $ EXPONENTIAL POLYNOMIALS .. 85 \ hspaceSince R∗{\ subf i N l l really}EXPONENTIAL depends now POLYNOMIALS only on k we shall\quad denote85 it R sub k from now on period Therefore comma from open parenthesis 8 closing parenthesis and open parenthesis 1 1 closing parenthesis we obtain \ hspaceEquation:∗{\ f open i l l } parenthesisSince $ 14 R closing{ N } parenthesis$ really .. dependsR sub k open now parenthesis only on x $ sub k 1 $ closing we shall parenthesis denote a sub it k comma$ R { j =k 0}$ if q fromplus j lessnow 0 on comma . Equation: open parenthesis 1 5 closing parenthesis .. R sub k open parenthesis x EXPONENTIAL POLYNOMIALS 85 sub 1 closing parenthesis a sub k comma j = b sub k comma q plus j if 0 less or equal q plus j less or equal q period Since R really depends now only on k we shall denote it R from now on . \noindentOne can introduceTherefore polynomials , from S (subN 8 k ) in and a similar ( 1 way 1 ) and we compute obtain explicitly k Therefore , from ( 8 ) and ( 1 1 ) we obtain S sub k open parenthesis e to the power of x sub 1 closing parenthesis a sub k comma j for the same values of j comma minus\ begin 2{ na less l i g orn ∗} equal j less or equal 0 period .. As a corollary one obtains \ tagProposition∗{$ ( 2 14 period ) 1 $ period} R Let{ k f in} E( sub n x comma{ 1 1} open) parenthesis a { k K closing , j parenthesis} = 0 .. and i f k in N q comma + Rk(x1)ak,j = 0 ifq + j < 0, (14) ..j there< exist0 non , hyphen\\\ tag zero∗{$ polyno ( hyphen1 5 ) $} R { k } ( x { 1 } ) a { k , j } = R (x )a = b if0 ≤ q + j ≤ q. (15) b mials{ k R sub , k q comma + S sub j } k ..iof f a single 0 k variable\1 leqk,j commaqk,q+j + .. with j coefficients\ leq q in K comma . N sub k in N comma .. \end{ a l i g n ∗} and positive x1 One can introduce polynomials Sk in a similar way and compute explicitly Sk(e )ak,j for constants Cthe sub same k comma values D of subj, − k2n ..≤ suchj ≤ that0. theAs adistributions corollary one a obtainssub k comma j comma minus 2 n less or equal j lessOne or can equal introduce 0 comma ..polynomials defined by the $ S { k }$ in a similar way and compute explicitly Proposition 2 . 1 . Let f ∈ En,1(K) and k ∈ N, there exist non - zero polyno $Laurent S { developmentk } ( e .. ˆinfinity{ x { 1 }} ) a { k , j }$ for the same values of $ j , - mials Rk, Sk of a single variable , with coefficients in K,Nk ∈ N, and − bar2 f bar n to the\ leq power ofj 2 lambda\ leq = sum0 a . sub $ k\ commaquad jAs open a parenthesis corollary lambda one obtains plus k closing parenthesis to the positive constants Ck,Dk such that the distributions ak,j, −2n ≤ j ≤ 0, defined by power of j the Laurent development ∞ Propositionsatisfy the estimates 2 . 1 .. . j Let = minus $ f2 n \ in E { n , 1 } ( K ) $ \quad and $ k \ in N , $ \quad there exist non − zero polyno − Equation: open parenthesis 1 6 closing parenthesis2λ ..X bar angbracketleftj R sub k open parenthesis x sub 1 closing | f | = ak,j(λ + k) parenthesismials $ a R sub{ k commak } ,S j comma{ phik right}$ angbracket\quad of bar a plus single bar angbracketleft variable , S\ subquad k openwith parenthesis coefficients e to the in power$ K of , x sub Nsatisfy 1 closing{ k the}\ parenthesis estimatesin N a subj = , k− $ comma2n\quad j commaand positivephi right angbracket bar less or equal C sub k bar phi bar Nconstants sub k in x to the $ C power{ k of} maximum,D supp{ openk }$ parenthesis\quad such phi closing that parenthesis the distributions e to the power of $ open a { parenthesisk , Dj sub} k, rho− open2 parenthesis n \ xleq closingj parenthesis\ leq closing0 parenthesis , $ \quad commadefined where by phi thein D open parenthesis C to the Laurent development \quad $ \ infty $x1 max (Dkρ(x)) power of n closing parenthesis| hRk(x comma1)ak,j, φ rhoi | + open| hS parenthesisk(e )ak,j, φ xi |≤closingCk k parenthesisφ k Nk ∈x =supp( log openφ)e parenthesis, (16) 1 plus bar x bar n closing parenthesis plus bar Re x sub 1 bar period whereφ ∈ D(C ), ρ(x) = log(1+ | x |)+ | to0 such the power that the of primeestimate D sub( 1 k 6 to ) theimplies power of prime greater 0 such that the estimate open parenthesis 1 6 closing parenthesis .. implies 0 \noindent satisfy the estimatesk \quad $0 j = −max 2 n(D $ kρ(x)) bar angbracketleft x m sub 1 to| the hxm power1 ak,j, of φi k |≤ a subCk k kφ commak Nk ∈ jx commasupp( phiφ)e right angbracket bar less or equal C sub k to the power of prime bar phi bar N sub k in x to the power of maximum supp open parenthesis phi closing parenthesis \ begin { a l i g n ∗}In reality , one needs these estimates for the distributions involved e to the powerin of open the analytic parenthesis continuation D sub k to of distributionthe power of - valuedprime rho holomorphic open parenthesis functions x of closing the form parenthesis closing parenthesis\ tag ∗{$ ( 1 6 ) $}\mid \ langle R { k } ( x { 1 } ) a { k , j } | f1 |2λ1 ... | f |2λp /(| f1 |2 +...+ | f |2)m. These functions have already appeared in , In\ ..phi reality\ commarangle .. one\p ..mid needs+ .. these\mid .. estimatesp \ langle .. for .. theS ..{ distributionsk } ( e.. involved ˆ{ x { .. in1 ..}} the) a { k , j } , our\phi previous\rangle work [ 2 ,\mid 1 0 ] .\ leq The existenceC { k of}\ an analyticparallel continuation\phi as a\ meroparallel - N { k } analytic continuationmorphic function of distribution of λ , ..., hyphen λ follows valued from holomorphic Hironaka ’ functions s resolution of the of singularities form , \ inbar{ f 1x bar}ˆ to{\ themax power} ofsupp 2 lambda1 ( subp\ 1phi period) period e period ˆ{ (D bar f sub{ pk bar}\ to therho power( of 2 x lambda ) sub ) p} slash, \\ where \phi \ in butD since ( we C want ˆ{ tn o} control), the distributions\rho ( that x appear ) as = coefficients\ log in( the 1Laurent + \mid open parenthesisdevelopments bar f 1 bar to about the somepower pole of 2 , plus that period i s , we period would period like to plus obtain bar estimates f sub p bar similar to the to power those of 2 closing parenthesisx \mid to the) power + of\mid m period\Re .... Thesex functions{ 1 }\ havemid already. appeared in \end{ a l i g n ∗}of Proposition 2 . 1 and Corollary 2 . 1 , we need t o find some kind of functional equation our previousthat work provides open square the analytic bracket continuation 2 comma .. . 1 0 Since closing it square i s easier bracket period .. The existence of an analytic continuation as a mero hyphen 2λ1 2λp Corollary 2to . provide 1 . \quad functionalI f equations $ K \ forsubseteq| f1 | ... |Qfp | ,, we $ need\quad a t echnicalther e \ trickquad t oare integers $ m { k } morphic functionreduce of this lambda kind sub of 1quotients comma period of functions period t period o products comma . lambda It is sub based p follows on a simple from Hironaka lemma quoteright s\ in resolutionN of , singularities $ \quad commaand two \quad constants $ C ˆ{\primeabout the} { inversek ˆ{ Mellin, }} transformD ˆ{\prime . In order} { tk o simplify} > it0 s $ writing such let that us introduce the estimate the ( 1 6 ) \quad i m p l i e s but since wefollowing want t o notation control the . distributions that appear as coefficients in the Laurent developments about some pole comma that i s comma we would like to obtain estimates For t1, ..., tp > 0, µ1, ..., µp ∈ C, we let \ [ similar\mid to those\ langle of Propositionx{ m } 2{ period1 }ˆ 1{ andk } Corollarya { 2k period , 1 comma j } , we need\phi t o find\rangle some kind \mid \ leq C ˆof{\ functionalprime } equation{ k }\ thatparallel provides the analytic\phi continuation\ parallel periodN .. Since{ k }\ it i sin easier{ x }ˆ{\max } supp ( \phi ) e ˆ{ ( D ˆ{\prime } { k }\rho ( x ) ) }\ ] to provide functional equations for bar f 1 bar to the power of 2 lambda sub 1∗ periodµ µ period1 µp period bar f sub p bar to t := t1 ...tp . the power of 2 lambda sub p comma we need a t echnical trick t o Givens , ..., s , β ∈ , let reduce this kind of quotients of functions t o products period .. It is based1 onp a− simple1 C In \quad r e a l i t y , \quad one \quad needs \quad these \quad e s t i m a t e s \quad f o r \quad the \quad distributions \quad involved \quad in \quad the lemma about the inverseds Mellin:= ds transform1...dsp−1, s periodp := β − Ins order1 − ... t− os simplifyp−1, µ˜j := itµ sj writing− sj(1 let≤ j us≤ p). analyticintroduce the continuation following notation of distribution period − valued holomorphic functions of the form For t sub 1 commaWe also period let s period:= (s period1, ..., sp comma−1), s˜ t:= sub p (s greater1, ..., sp) 0, commawith sp muas previously 1 comma period defined period . period comma mu\noindent sub p in C comma$ \mid we letf 1 \mid ˆ{ 2 \lambda { 1 }} ... \mid f { p }\mid ˆ{ 2 \lambdat to the power{ p }} of * mu/( : = t sub\mid 1 to thef power 1 of mu\mid 1 periodˆ{ 2 period} + period . t sub . p to . the + power\mid of mu subf p{ periodp } Given\mid sˆ{ sub2 1} comma) ˆ{ periodm } period. $ period\ h f i comma l l These s sub functions p minus 1 comma have beta already in C comma appeared let ds in : = ds sub 1 period period period ds sub p minus 1 comma s sub p : = beta minus s sub 1 minus period period period minus s sub p minus 1\noindent comma mu-tildeour sub previous j : = mu work sub j minus [ 2 , s\ subquad j open1 0 parenthesis ] . \quad 1 lessThe or equal existence j less or ofequal an p closinganalytic parenthesis continuation as a mero − periodmorphic function of $ \lambda { 1 } ,..., \lambda { p }$ follows from Hironaka ’ s resolution of singularities , We also let s : = open parenthesis s sub 1 comma period period period comma s sub p minus 1 closing parenthesis comma\noindent s-tildebut : = open since parenthesis we want s sub t o 1 comma control period the period distributions period comma that s sub appearp closing asparenthesis coefficients comma with in the sLaurent sub p as previously developments defined period about some pole , that i s , we would like to obtain estimates similar to those of Proposition 2 . 1 and Corollary 2 . 1 , we need t o find some kind of functional equation that provides the analytic continuation . \quad Since it i s easier

\noindent to provide functional equations for $ \mid f 1 \mid ˆ{ 2 \lambda { 1 }} ... \mid f { p }\mid ˆ{ 2 \lambda { p }} , $ we need a t echnical trick t o

\noindent reduce this kind of quotients of functions t o products . \quad It is based on a simple lemma about the inverse Mellin transform . In order t o simplify it s writing let us introduce the following notation .

\ centerline {For $ t { 1 } , . . . , t { p } > 0 , \mu 1 , . .., \mu { p }\ in C , $ we l e t }

\ begin { a l i g n ∗} t ˆ{ ∗ \mu } : = t ˆ{\mu 1 } { 1 } . . . t ˆ{\mu { p }} { p } . \\ Given s { 1 } , . . . , s { p − 1 } , \beta \ in C , l e t \\ ds : = ds { 1 } . . . ds { p − 1 } , s { p } : = \beta − s { 1 } − ... − s { p − 1 } , \ tilde {\mu} { j } : = \mu { j } − s { j } ( 1 \ leq j \ leq p ) . \end{ a l i g n ∗}

\noindent Wealsolet $s : = ( s { 1 } , . . . , s { p − 1 } ), \ tilde { s } : = ( s { 1 } , . . . , s { p } ) , $ with $ s { p }$ as previously defined . 86 .. C period A period BERENSTEIN AND A period YGER \noindentRecall also86 the\ somewhatquad C . standard A . BERENSTEIN notation comma AND A . YGER Capital Gamma open square bracket a closing square bracket : = Capital Gamma open square bracket a sub 1 comma\noindent periodRecall period period also comma the somewhat a sub k closing standard square bracket notation : = Capital , Gamma open parenthesis a sub 1 closing parenthesis period period period Capital Gamma open parenthesis a sub k closing parenthesis comma 86 C . A . BERENSTEIN AND A . YGER \ [ for\Gamma complex values[ a a sub ] j such : that = the\ EulerGamma Gamma[ function a { i1 s defined} , period . Finally . . comma , as a { k } ] Recall also the somewhat standard notation , :long = as\ thereGamma i s no possibility( a { of1 confusion} )... comma we shall use\Gamma the following( abbreviated a { k } ), \ ] notation for multiple integrals on lines parallel to the imaginary axes period .. Let gamma = Γ[a] := Γ[a , ..., a ] := Γ(a )...Γ(a ), open parenthesis gamma 1 comma period period period1 commak gamma1 subk p minus 1 closing parenthesis be a vector of\noindent real componentsforfor complex complex comma then values values commaaj such for $ that aany{ the integrablej Euler}$ Gamma functionsuch that function F comma the i s Euler defined Gamma . Finally function , as long as i s defined . Finally , as longgamma as plus therethere i infinity i i s s nogamma no possibility possibility 1 plus of confusioni infinity of gamma confusion, we shall sub use p ,minus the we following shall 1 to the abbreviated use power the of plus following i infinity abbreviated integral F open parenthesis s closingnotation parenthesis for multiple ds integrals: = integral on lines period parallel period to theperiod imaginary integral axes F open . Let parenthesisγ = (γ1, ..., s closing γp−1) parenthesis ds\noindent sub 1 periodnotationbe period a vector period for of real ds multiple sub components p minus integrals , 1 then gamma , for minus on any lines integrable i infinity parallel function gamma 1F, to minus the i infinityimaginary gamma axes sub p . minus\quad Let 1$ minus\gamma i infinity= $ Equation: C open square bracket mu 1 comma period period period comma mu sub p closing square $ ( \gamma 1 , . . . , \gamma { p − } 1 ) $ be a vector of real components , then , for any integrable function bracket comma .. Lemma 2 period 2 period Let t sub 1 commaγ + periodi∞ γ period1 + i∞ periodγ 1 comma+i∞ t sub p greater 0 comma mu $ F , $ p− 1 comma period period period comma muZ sub p in C commaZ Re betaZ greater 1 comma P in then comma with the previous notation commaF (s)ds := ... F (s)ds1...dsp−1 \ beginEquation:{ a l i g open n ∗} parenthesis 1 7 closing parenthesis .. P open parenthesis mu closing parenthesis open parenthesis t sub\gamma 1 plus t period+ iperiod\ infty to the power\gamma of * sub period1 + to theγ − i poweri∞\ inftyγ of1 − mui∞ plus\γgammap t− sub1 − pi∞ closing{ p parenthesis− } 1 ˆ to{ the+ power i \ infty }\\\ int F ( s ) ds : = \ int ... \ int F ( s ) of beta = open parenthesisLemma 22 i. pi2. closing Lett1, parenthesis ..., tp > 0, to µ1 the, ..., power µp ∈ of 1C minus, <β > p 11 Capital,P ∈ GammaC[µ open1, ..., parenthesis µp], beta ds { 1 } . . . ds { p − 1 }\\\gamma − i \ infty \gamma 1 − i closing parenthesisthen integral, with theCapital previous Gamma notation from gamma, minus i infinity to gamma plus i infinity open square bracket s-tilde\ infty closing\gamma square bracket{ p P− open } parenthesis1 − tilde-mui \ infty closing\\ parenthesisLemma t to2 the . power 2 of * . mu-tilde Let ds t { 1 } , . . . , t { p } > 0 , \mu 1 , . . . , \mu { p }\ in for any gamma sub j greater 0 such that gamma 1 plus period periodZ period plus gamma sub p minus 1 less Re beta C, \Re \beta > 1∗ ,µ P β \ in \ tag1−p∗{1 $ C [ γ−i∞\mu 1∗µ˜ , . . . , minus 1 period P (µ)(t1 + t... + tp) = (2iπ) Γ(β) Γγ+i∞[˜s]P (˜µ)t ds (17) \muWe{ applyp } this] lemma , $ t} o the study of the coefficients in the Laurent e-line xpansion \end{ a l i g n ∗} about mu =for 0 of any theγ analyticj > 0 such continuation that γ1 + of... + γp−1 < <β − 1. Equation: open parenthesisWe apply this 1 8 lemma closing t parenthesis o the study .. of mu the arrowright-mapstocoefficients in the Laurent bar f bar e − toline the xpansion power of * 2 open parenthesis\noindent muthenabout t minus ,µ with= k closing 0 of the the parenthesis analytic previous continuation slash notation bar f barof to, the power of 2 m comma where t in closing square bracket 0 comma infinity open square bracket to the power of p is a vector t o be chosen \ begin { a l i g n ∗} below comma mu in C comma k in Z comma k iµ s7→| thef p|∗ hyphen2(µt−k) dimen/ k f k hyphen2m, (18) \ tagsional∗{$ vec ( open 1 parenthesis 7 ) $ k} commaP( period\mu period) period ( comma t { k1 closing} + parenthesis t { .. comma} mˆ{ in ∗ N } toˆ{\ themu power} { . } p +of * comma t { fp sub}where j in) Etˆ{\∈ sub]0,beta∞ n[ commais} a vector= 1 open t ( o parenthesis be 2 chosen i Kbelow{\ closingpi, µ ∈} parenthesisC,) k ∈ ˆ{Z,1 k commai s{ the − barp−{ dimen fp bar}} 2 - m1 =}\ openGamma parenthesis( bar\beta f 1 bar) to the\ int power of\Gamma 2 plus periodˆ{\gamma period period− plusi bar\ finfty sub p bar} {\ to thegamma power of+ 2 Case i 1 m\ infty Case 2} comma[ \ tilde { s } ]P( \ tilde {\mu} ) t ˆ{ ∗ \ tilde {\mu}} ds and comma .. keeping with the previous∗ notation comma bar f bar to the power2 of * r = bar2 f 1 openm barRow 1 r sub \end{ a l i g n ∗} sional~(k, ..., k), m ∈ N , fj ∈ En,1(K), k f k 2m = (| f1 | +...+ | fp | ) 1 Row 2 period period period closing bar. f sub p bar to the power of r p .. for any vector , r = open parenthesis r sub 1 comma period period period comma r sub p closing parenthesis open parenthesis similar \noindent f o r any $ \gamma { j } > 0 $ such that $ \gamma 1 + . . . + meaning for f to the power of * r closing parenthesis period ∗r r1 r and , keeping with the previous notation , | f | = | f1 | | fp | p for any \gammaFrom Proposition{ p − ..} 11 period< 2 we\Re conclude\beta that there− i s1 a polynomial . $ A open parenthesis... lambda comma x sub 1 ∗r closing parenthesisvector .. andr = (r1, ..., rp)( similar meaning for f ). \ hspace ∗{\ f i l l }We apply this lemma t o the study of the coefficients in the Laurent $ e−l i n e $ differential operatorsFrom Proposition Q 1 comma j to 1 . the 2 we power conclude of open that parenthesis there i s a polynomial lambda subA( commaλ, x1) x and comma differential e to the power of x xpansion (λ x1 −x1 sub 1 comma eoperators to the powerQ1, of j, minusx, e , x e sub, ∂/∂x1 comma) such partialdiff that slash partialdiff x closing parenthesis such that A open parenthesis lambda comma x sub 1 closing parenthesis f 1 to the power of lambda sub 1 period period period λ +1 \noindent about $ \mu = 0 $ ofλ1 theλp analytic(λ)(f continuationλ1 j λp of f sub p to the power of lambda sub pA =(λ, Q x 11) commaf1 ...f jp to= theQ1 power, j of1 open...fj parenthesis...fp ). lambda closing parenthesis open parenthesis f 1One to the can power iterate of this lambda functional sub 1 periodequation period and restrict period f our sub attention j to the t power o λ of= lambdaµt. After sub some j plus 1 period \ begin { a l i g n ∗} period period fwork sub p one to the obtains power the of following lambda sub result p closing . parenthesis period \ tag ∗{$ ( 1 8 ) $}\mu \mapsto \mid f \mid ˆ{ ∗ 2 ( \mu t − k One can iterate thisProposition functional equation2 . 2 . andLet restrictf1, ..., our f ∈ attentionE ( ), t othen lambda, for = anymu tt period∈]0, 1[p( outside ) } / \ parallel f \ parallel ˆ{ 2 mp } n,,1 K After some worka countable one obtains union the following of K− algebraic result period hypersurfaces , which depend on the fj) and \end{ a l i g n ∗} ∗ Propositionany 2 periodk 2∈ periodZ, m Let f∈ 1 commaN , periodthere period are periodpolynomials commaR fk suband p in ES subk nin commaK[u] 1 openand parenthesis K closing parenthesisconstants commaC ,D .. then> comma0,N for∈ any t insuch closing thatsquare i line bracket−fa 0 comma∈ 1D open0( n) squaredenote bracket to the \noindent where $ t k \k in ]k 0 , N \ infty [ ˆ{ p }$k,j is a vectorC t o be chosen below power of p openthe parenthesis coefficients outside of the Laurent expansion $ ,a ....\ countablemu \ in .... unionC .... , of Kk hyphen\ in algebraicZ , .... hypersurfaces k$ isthe comma $p .... which− $ .... dimen depend− .... on .... the .... f sub j closing parenthesis .... and j=−2n \ [ \anyl e f k t in.Z s i comma o n a l m\ invec N{} to the(k,...,k),m power of * comma .. there .. are polynomials∗ 2(µt−k) R subX k .. and\jin S subN k .. ˆ{ in ∗ K } open, | f |k f k2m = ak,jµ , squaref { bracketj }\ uin closingE square{ n bracket , 1.. and} ..(K), constants \ parallel f \ parallel 2 m = ( \mid f 1 \mid ˆ{ 2 } + . . . + \mid f { p∞}\mid ˆ{ 2 } )\ begin { a l i g n e d } & C sub k comma D sub k greater 0 comma N sub k in N .. such .. that i line-f a sub k comman j in D to the power of then, for − 2n ≤ j ≤ 0, φ ∈ D(C ), mprime\\ open parenthesis C to the power of n closing parenthesis .. denote .. the .. coefficients .. of the x1 max (Dkρ(x)) &,Laurent\ expansionend{ a l i g n| e hR d }\k(xright1)ak,j,. φ\i] | + | hSk(e )ak,j, φi |≤ Ck k φ k Nk ∈x supp(φ)e , bar f bar sub bar to the power of * sub f to the power of 2 openwhereρ parenthesis(x) = log(1+ sub bar| x to|)+ the| < powerx1 | . of 2 m to the power of mu t minus k closing parenthesis = sum from j = minus 2 n to infinity a sub k comma j mu to the power of j comma then\noindent comma forand minus , \quad 2 n lesskeeping or equal j with less or the equal previous 0 comma phi notation in D open $ parenthesis , \mid C tof the power\mid ofˆ{ n closing∗ r } =parenthesis\mid commaf bar 1 angbracketleft\mid\ begin { Rarray sub k}{ openc} parenthesisr { 1 }\\ x sub... 1 closing parenthesis\end{ array a sub}\ kmid commaf j comma{ p } phi right\mid angbracketˆ{ r } barp $ plus\ barquad angbracketleftfor any vector S sub k open parenthesis e to the power of x sub 1 closing parenthesis a sub k comma$ r j = comma ( phi r right{ 1 angbracket} , bar . less . or .equal , C sub r k{ barp phi} bar) N ( sub $ k in similar x to the meaning power of maximum for $ f ˆ{ ∗ suppr } open) parenthesis . $ phi closing parenthesis e to the power of open parenthesis D sub k rho open parenthesis x closing parenthesis closing parenthesis comma where rho open parenthesis x closing parenthesis = log open parenthesis 1 plus barFrom x bar Proposition closing parenthesis\quad plus1 bar . 2 Re we x sub conclude 1 bar period that there i s a polynomial $A ( \lambda , x { 1 } ) $ \quad and differential operators $Q 1 , j ˆ{ ( \lambda } { , } x , e ˆ{ x { 1 }} , e ˆ{ − x { 1 }} , \ partial / \ partial x )$ such that

\ [A( \lambda , x { 1 } ) f 1 ˆ{\lambda { 1 }} . . . f { p }ˆ{\lambda { p }} = Q 1 , j ˆ{ ( \lambda ) ( f } 1 ˆ{\lambda { 1 }} . . . f { j }ˆ{\lambda { j } + 1 } . . . f { p }ˆ{\lambda { p }} ). \ ]

\noindent One can iterate this functional equation and restrict our attention t o $ \lambda = \mu t . $ After some work one obtains the following result .

\ hspace ∗{\ f i l l } Proposition2.2.Let $f 1 , . . . , f { p }\ in E { n , 1 } ( K ) , $ \quad then , for any $ t \ in ] 0 , 1 [ ˆ{ p } ( $ o u t s i d e

\noindent a \ h f i l l countable \ h f i l l union \ h f i l l o f $ K − $ a l g e b r a i c \ h f i l l hypersurfaces , \ h f i l l which \ h f i l l depend \ h f i l l on \ h f i l l the \ h f i l l $ f { j } ) $ \ h f i l l and

\noindent any $ k \ in Z , m \ in N ˆ{ ∗ } , $ \quad the re \quad are polynomials $ R { k }$ \quad and $ S { k }$ \quad in $ K [ u ] $ \quad and \quad constants $ C { k } ,D { k } > 0 , N { k }\ in N $ \quad such \quad that i $ l i n e −f a { k , j }\ in D ˆ{\prime } ( C ˆ{ n } ) $ \quad denote \quad the \quad coefficients \quad o f the Laurent expansion

\ begin { a l i g n ∗} \mid f \mid ˆ{ ∗ } {\ parallel }ˆ{ 2 ( } { f }ˆ{\mu t − k ) } {\ parallel ˆ{ 2 m }} = \sum ˆ{ j = − 2 n } {\ infty } a { k , j }\mu ˆ{ j } , \\ then , f o r − 2 n \ leq j \ leq 0 , \phi \ in D ( C ˆ{ n } ), \\\mid \ langle R { k } ( x { 1 } ) a { k , j } , \phi \rangle \mid + \mid \ langle S { k } ( e ˆ{ x { 1 }} ) a { k , j } , \phi \rangle \mid \ leq C { k }\ parallel \phi \ parallel N { k }\ in { x }ˆ{\max } supp ( \phi ) e ˆ{ (D { k }\rho ( x ) ) } , \\ where \rho ( x ) = \ log ( 1 + \mid x \mid ) + \mid \Re x { 1 }\mid . \end{ a l i g n ∗} EXPONENTIAL POLYNOMIALS .. 87 \ hspaceCorollary∗{\ 2f i period l l }EXPONENTIAL 2 period If K POLYNOMIALS subset equal Q\ commaquad 87 there is an integer nu sub k in N comma .. and positive constants \ hspaceC sub∗{\ k tof the i l l power} Corollary of comma 2 to . the 2 power . If of $K prime D\subseteq sub k to the powerQ of , prime $ there such that is an integer $ \nu { k } \ inbar angbracketleftN , $ \quad x nu suband 1 to positive the power ofconstants k a sub k comma j comma phi right angbracket bar less or equal C sub EXPONENTIAL POLYNOMIALS 87 k to the power of prime bar phi bar N sub k in x to the power of maximum supp open parenthesis phi closing parenthesis \noindent $ C ˆCorollary{\prime } 2{ . 2k . ˆIf{ K, ⊆}}Q, Dthere ˆ{\ is anprime integer} {νkk∈ }N$, and such positive that constants e to the power of0 open0 parenthesis D sub k to the power of prime rho open parenthesis x closing parenthesis closing C , D such that parenthesis periodk k \ [ \mid \ langle x{\nu } { 1 }ˆ{ k } a { k , j } , \phi \rangle \mid \ leq 3 period .. Localization of ideals andk applications0 period .. Inmax open square( bracketD0 ρ(x)) 7 closing square bracket we gave | hxν a , φi |≤ C k φ k N ∈ supp(φ)e k . Csome ˆ{\ sufficientprime } { k }\ parallel1 k,j\phi k\ parallelk x N { k }\ in { x }ˆ{\max } supp ( conditions\phi comma)3 e . albeit ˆ{ Localization( sometimes D ˆ{\ hardprime of ideals t o verify} and{ k comma applications}\ sorho that if( . f 1 comma xIn [ 7 ) ] period we ) gave} period some. \ period] sufficient comma f sub n are exponentialconditions , albeit sometimes hard t o verify , so that if f1, ..., fn are exponential polynomialspolynomials in n variables in withn variables integral with frequencies integral whosefrequencies variety whose of common variety of common zeros V = {z ∈ 3 . \quad Localizationn of ideals and applications . \quad In [ 7 ] we gave some sufficient zeros V = openC : bracef1( zz) in = C... to= thefn( powerz) = 0 of} is n :discrete f 1 open or parenthesis empty , then z closing the ideal parenthesisI generated = period by them period in period = f conditions , albeit sometimesn hard t o verify , so that if $ f 1 , . . . , sub n open parenthesisthe space z closingAρ(C ), parenthesis ρ(z) = log =(1+ 0 closing| z |)+ | brace

What can one say when we do not assume the ideal i s either complete intersec − tion or its variety i s discrete $ ? $ \quad There are several ideals containing $ I =I(f1,...,f { p } ) . $

\noindent First , let us recall $ that \surd{ I } { , }$ the radical of $ I , $ i s the set of all elements $ F \ in A {\rho } ( C ˆ{ n } ) $ such that $ F ˆ{ k }\ in I $ for some $k \ in N . $ \quad Second , let $ \hat{ I } { , }$ the lo cal integral closure of $ I , $ be the set of all elements $ F \ in A {\rho } ( C ˆ{ n } ) $ such that for every point $ x { 0 }\ in C ˆ{ n }$ ther e i s 88 .. C period A period BERENSTEIN AND A period YGER \noindenta neighborhood88 \quad U andC a constant . A . BERENSTEIN C sub x sub 0 AND greater A 0 . such YGER that Line 1 p Line 2 bar F open parenthesis x closing parenthesis bar less or equal C sub x sub 0 bar f open parenthesis x closing\noindent parenthesisa neighborhood bar = C sub x sub $ U 0 $open and parenthesis a constant sum bar $ f sub C j{ openx parenthesis{ 0 }} > x closing0 $ parenthesis such that bar to the power of 2 closing parenthesis to the power of 1 slash 2 comma forall x in U period Line 3 j = 1 88 C . A . BERENSTEIN AND A . YGER \ [ \Forbegin W open{ a l i gin n C e d to} thep \\ power of n comma let I sub W denote the ideal generated by f 1 comma period period period a neighborhood U and a constant C > 0 such that comma\mid f subF p in (H open x parenthesis ) \mid W closing\ leq parenthesisx0C { x period{ 0 .. It}}\ parallel f ( x ) \ parallel = C { x { 0 }} ( \sum \mid f { j } ( x ) \mid ˆ{ 2 } ) ˆ{ 1 / 2 } follows from .. open square bracket 22 closing square bracket .. that F in I-circumflexp if and only if for every x sub 0 , \ f o r a l l x \ in U. \\ in C to the power of n there i s an open X j = 1 \end{ a l i g n e d }\| F](x) |≤ C k f(x) k = C ( | f (x) |2)1/2, ∀x ∈ U. neighborhood W comma a positive integerx0 N comma and functionsx0 j phi 1 comma period period period comma phi N such that j = 1 F to the power of N plus phi 1 to the power of F to the power of N minus 1 plus period period period plus phi N = 0 n in\noindent W comma andForFor phiW $W$ subopen j in in openIC sub, let W inIW todenote the $Cˆ power the{ n of ideal} period generated, $ to the l e tby powerf $1, ..., Iof fj p{ inWH}($W ). denoteIt follows the from ideal generated by $ f 1 , . . .ˆ , f { p }$ in $ Hn ( W ) . $ \quad I t Finally comma[ 22 let ] I that openF parenthesis∈ I if and V only closing if for parenthesis every x0 =∈ openC there brace i Fs an in openA sub neighborhood rho open parenthesisW, a C to the powerfollows of n closing frompositive parenthesis\quad integer[ 22 :N, F bar]and\quad V functions = 0that closingφ1, ..., brace $ φN F periodsuch\ in that Note\hat that{ line-fI } $ sub if or and a function only F if to b for elong every $ xto I{ sub0 loc}\ meansin thatC itˆ{ vanishesn }$ on there the points i s of an the open variety V with some multiplicity comma neighborhood $W , $ aN positiveF N−1 integer $N , $ and functionsj $ \phi 1 , whereas in I open parenthesis VF closing+ φ1 parenthesis+ ... + ..φN the= common 0 inW, multiplicitiesand φj ∈ I ofW . f 1 comma period period period ..., \phi N $ such that comma f sub p .. are disregarded period .. It in s Finally , let I(V ) = {F ∈ Aρ(C ): F | V = 0}. Note that line − for a function F to b elong obvious that I open parenthesis V closing parenthesis i s a closed ideal comma and we recall that the same is true for to Iloc means that it vanishes on the points of the variety V with some multiplicity , whereas I\ sub[ F loc ˆ{ periodN } + \phi 1 ˆ{ F ˆ{ N − 1 }} + . . . + \phi N = 0 in in I(V ) the common multiplicities of f1, ..., fp are disregarded . It i s WSome , inclusions and \ bphi etween{ thesej }\ idealsin are clearI ˆ{ : j } { W ˆ{ . }}\] obvious that I(V ) i s a closed ideal , and we recall that the same is true for Iloc. Some I subset equalinclusions I sub loc b subsetetween equal these I-circumflex ideals are clear subset : equal I open parenthesis V closing parenthesis comma surd of I subset equal I open parenthesis to the power of line-V closing parenthesis period \noindent Finally,let $I ( V ) = \{ F \ in A {\rho } ( C ˆ{ n } It is also clear that comma in general comma we do not have√ I sub locline− =V I open parenthesis V closing parenthesis I ⊆ Iloc ⊆ Iˆ ⊆ I(V ), I ⊆ I( ). period):F We are now\mid readyV t o = 0 \} . $ Note that $ line −f { or }$ a function $F$ to b elong state two importantIt is also results clear that period , in general , we do not have Iloc = I(V ). We are now ready t o toTheorem $ I 3{ periodstatel o c two} 2$ period important means Let I that results be the it . ideal vanishes in A sub on rho theopen pointsparenthesis of C the to the variety power of n $ closing V $ parenthesis with some multiplicity , whereasin $I ( V )$ \quad thecommon multiplicitiesn of $ f 1 , . . generated by f 1 commaTheorem period period3 . period 2 . Let commaI be the f sub ideal p in in EA subρ(C n comma) generated 1 open by parenthesisf1, ..., fp ∈ CEn, closing1(C), parenthesis comma. , f { p }$ \quad are disregarded . \quad I t i s Equation: V = open brace x in C to the power of n : f 1 open parenthesis x closing parenthesis = period period period √ n =\noindent f sub p openobvious parenthesis that x closing $T hen I parenthesisI (= I( VV =).V 0 )closing $ i brace s a period closed= ..{x Then∈ idealC surd: f1( , ofx and) I = =... I we open= f recallp( parenthesisx) = 0} that. V closing the same is true for parenthesis$ I { l o period c } . $ Theorem 3 . 3 . Let I be the ideal of the previous theorem and let m be given by m = SomeTheorem inclusions 3 period 3 b period etween Let I these be the ideal ideals of the are previous clear theorem : and let m be given inf (p + 1, n), then Iˆ2m ⊆ I. by m = inf open parenthesis p plus 1 comma n closing parenthesis comma then I-circumflex to the power of 2 m subset One of the remarkable consequences of Theorem 3 . 3 i s the conclusion that sometimes equal\ [I I period\subseteq I { l o c }\subseteq \hat{ I }\subseteq I(V), \surd{ I } the variety of common zeros i s interp olating ( see [ 4 , 5 ] for background information on \subseteqOne of the remarkableI ( ˆ{ consequencesl i n e −V } of). Theorem\ ] 3 period 3 i s the conclusion that this question . ) sometimes the variety of common zeros i s interp olating open parenthesis see open square bracket 4 comma 5 closing Proposition 3 . 1 . Let f1, ..., f ∈ E ( ) be such that dim V = 0 and J(x) 6= 0 for square bracket for background n n,1 C every x ∈ V, where J is the Jacobian determinant of the f . Then there is a constant \noindentinformationIt on is this also question clear period that closing , in parenthesis general , we do notj have $ I { l o c } = I ( C > 0 such that VProposition ) . $ 3 period Wearenowready 1 period Let f 1 comma to period period period comma f sub n in E sub n comma 1 open parenthesis C closing parenthesis be such that dim V = 0 and J open parenthesis x closing parenthesis equal-negationslash \noindent0 for everystate x in V commatwo important where J is the results Jacobian . determinant of the f sub j period .. Then there is | J(x) |≥ exp(−C(| 0 $ such that

\ begin { a l i g n ∗} \ tag ∗{$ ( 1 9 ) $}\mid J ( x ) \mid \geq \exp ( − C( \mid \Re x { 1 }\mid + \ log ( 2 + \mid x \mid ))) \ f o r a l l x \ in V. \end{ a l i g n ∗}

\noindent Thus $ , V $ \ h f i l l i s \ h f i l l an \ h f i l l interpolating \ h f i l l v a r i e t y \ h f i l l in \ h f i l l the \ h f i l l space $ A {\sigma } ( C ˆ{ n } ) $ \ h f i l l f o r any \ h f i l l weight $ \sigma \geq $

\ begin { a l i g n ∗} \mid \Re x { 1 }\mid + \ log ( 2 + \mid x \mid ). \end{ a l i g n ∗}

In f a c t , \quad one has \quad a \quad stronger result . \quad Let \quad $ f 1 , . . . , f { p }\ in E { n , 1 } ( C ) $ \quad b e such that dim $V = k$ and assume that , \quad at every point $ x \ in V , $ \quad the re i s a $ k \times k $ minor o f theJacobianmatrix $Df$ of $f 1 , . . . , f { p } , $ which does not vanish . \quad Then , the variety

\noindent $ V $ i s an interpolation variety for any weight $ \geq \mid \Re x { 1 } \mid + $ l o g $ ( 2 + \mid x \mid ) . $ Namely , if we l e t $ J { 1 } ,...,J { l }$ denote all the $ k \times k $ minors o f $ Df , $ then

\ [ \mid J { 1 } ( x ) \mid + . . . + \mid J { l } ( x ) \mid \geq \exp ( − C( \mid \Re x { 1 }\mid + \ log ( 2 + \mid x \mid ))). \ ]

\noindent From [ 5 , Theorem 1 ] , one obtains that $ V $ i s an interp olating variety .

We conclude this short summary of some of our recent results with an indica − tion of some simple applications to harmonic analysis that can b e obtained from EXPONENTIAL POLYNOMIALS .. 89 \ hspacethe above∗{\ statementsf i l l }EXPONENTIAL and the methods POLYNOMIALS of open square\quad bracket89 4 closing square bracket period .... For that purpose comma let us recall that \noindenta linear differentialthe above operator statements P open parenthesis and the D methods closing parenthesis of [ 4 with ] . constant\ h f i l l coefficientsFor that and purpose commensurable , let us recall that time lags i s a finite sum of the form EXPONENTIAL POLYNOMIALS 89 \noindentLine 1 opena parenthesis linear differential 20 closing parenthesis operator open parenthesis $ P ( P open D parenthesis ) $ with D closing constant parenthesis coefficients phi closing and commensurable the above statements and the methods of [ 4 ] . For that purpose , let us recall that parenthesistime lags open i parenthesis s a finite t comma sum of x closing the form parenthesis = sum p sub j k open parenthesis D to the power of j phi a linear differential operator P (D) with constant coefficients and commensurable time lags i closing parenthesis open parenthesis t minus kT comma x closing parenthesis comma Line 2 t in R comma x in R to the s a finite sum of the form power\ [ \ begin of n{ commaa l i g n e open d } ( parenthesis 20 n ) greater ( equal P 0 ( closing D parenthesis ) \phi comma) D = ( open t parenthesis , x partialdiff ) = slash\sum partialdiffp { j } t commak ( partialdiff D ˆ{ slashj }\ partialdiffphi x) sub 1 ( comma t period− kT period ,period x comma ) partialdiff , \\ slash partialdiff x subt n closing\ in parenthesisR , comma x \ jin in N toR the ˆ{ powern } of, n plus ( 1 comma n \geq k inX Z comma0j ) T greater , D 0 comma = ( \ partial (20) (P (D)φ)(t, x) = pjk(D φ)(t − kT, x), / and\ partial p sub j k in Ct period , ..\ Thepartial symbol of/ this operator\ partial P openx parenthesis{ 1 } tau,..., comma xi closing parenthesis\ partial is the n n+1 element/ \ partial of E sub n plusx t ∈ 1{ commaRn, x}∈ R 1) open, ( ,n parenthesis≥ 0) j ,D =\ in ( C∂/∂t, closingN ∂/∂x ˆ{ parenthesis1,n ..., ∂/∂x +n), 1 j ∈} N ,, k k∈ Z, T\ in > 0, Z,T > 0given , \ byend{ a l i g n e d }\ ] and p k ∈ . The symbol of this operator P (τ, ξ) is the element of E ( ) given by Equation: open parenthesisj C 2 1 closing parenthesis .. P open parenthesis tau comman xi+1, closing1 C parenthesis : = e to the power of i open parenthesis t tau plus x times xi closing parenthesis P open parenthesis D closing parenthesis e to \noindent and $ p { j } k \ in C . $ \quad XThe symbol of this operator $ P ( the power of minus i open parenthesisP (τ, ξ t) tau := e plusi(tτ+ x·ξ times)P (D) xie− closingi(tτ+x·ξ parenthesis) = p k( =−iζ sum)jeikT p sub τ , j k open parenthesis(21) minus i zeta\tau closing, parenthesis\ xi ) to $ the is power the of element j e to the power of $E of i kT{ taun comma +j 1 , 1 } ( C ) $ given by with zeta =with openζ parenthesis= (τ, ξ). ( tau By the comma introduction xi closing of parenthesis the new coordinate period openξ0 = parenthesisiT τ, we are By in the the introductioncase of of the new coordinateexponential xi 0 = iT tau polynomials comma we considered are in at the beginning of this section . ) \ beginthe case{ a l ofi g exponentialn ∗} polynomials considered at the beginning of this section period closing parenthesis \ tag ∗{$ ( 2Theorem 1 ) $3} .P( 4 . Let P1\(Dtau), ..., Pn,+1(D\)xibe differential) : operators = e with ˆ{ i time lags( tas in \tau + Theorem 3 period( 20 ) , 4with period the Let property P sub that 1 open the characteristic parenthesis D variety closing parenthesis comma period period period comma Px sub\ ncdot plus 1 open\ xi parenthesis) } P D closing ( D parenthesis ) e be ˆ{ differential − i operators ( t with\tau time lags+ x \cdot \ xi ) } = \sum p { j } k ( − i \zeta ) ˆ{ j } e ˆ{ i kT \tau } , as in open parenthesis 20 closing parenthesisV := {ζ ∈ comman+1 : withP ( theζ) = property 0, 1 ≤ l that≤ n + the 1} characteristic variety \endV{ :a = l iopen g n ∗} brace zeta in C to the power of n plusC 1 : P subl l open parenthesis zeta closing parenthesis = 0 comma 1 less or equal l lessis discrete or equal and n plus all 1the closing points brace of V are simple . Then , every solution φ ∈ E(Rn+1)( \noindent with $ \zeta0 n+1= ( \tau , \ xi ) . ( $ By the introduction of the new coordinate is discrete andresp all., the φ ∈ points D (R of V)) areof simple the overdetermined period .. Then system comma every solution phi in E open parenthesis R to the power$ \ xi of n0 plus 1= closing iT parenthesis\tau , $ we are in theopen case parenthesis of exponential resp period comma polynomials phi in D considered to the power of at prime the open beginning parenthesis of R this to the section power of . n )plus 1 closing parenthesis closing parenthesis .. of theP overdetermined1(D)φ = ... = P systemn+1(D)φ = 0 (22) TheoremEquation: 3 . open 4 . parenthesis Let $P 22 closing{ 1 } parenthesis(D),...,P .. P sub 1 open parenthesis D closing{ parenthesisn + 1phi} =(D period can be represented in a unique way in the form of a series of exponential solutions of the period) $ beperiod differential = P sub n plus operators 1 open parenthesis with Dtime closing lags parenthesis phi = 0 system ( 22 ) , namely , ascan in be ( represented 20 ) , with in a unique the property way in the form that of the a series characteristic of exponential solutions variety of the system open parenthesis 22 closing parenthesis commaX namely comma φ(t, x) = c ei(tτ+x·ξ). \ [Line V 1 : phiopen = parenthesis\{\zeta t comma\ in x closingC ˆ{ parenthesisn + =ζ 1 sum} c:P sub zeta{ el to} the( power\zeta of i open) parenthesis = 0 t tau, plus 1 x times\ leq xi closingl \ parenthesisleq n period + Line 1 2\}\ zeta in] V ζ ∈ V

This s eries is convergent in the topology of E open parenthesisn+1 R to the power0 ofn+1 n plus 1 closing parenthesis open parenthesis respThis period s eries comma is convergent D to the power in the of topology prime openof E( parenthesisR )( resp R to., theD ( powerR )) of. n plus 1 closing parenthesis closing\noindent parenthesisis discrete period and all the pointsReferences of $ V $ are simple . \quad Then , every solution $ \Referencesphi \ in [ 1E ] ( M . F R . Aˆ{ t in y ah + , Resolution 1 } ) of $singularities and division of distributions , Comm . (open resp square $Pure . bracket , Appl 1\ closing.phi Math .square 23\ in ( 1 bracket 9 70D ) ˆ ,{\ 145 .. Mprime – 1period 50 . } F period( R A ˆ t{ i yn ah comma+ 1 Resolution} ) ) of $ singularities\quad of and the overdetermined system division of distributions[2] comma C.A.Berenstein,R.GayandA.Yger, Comm period .. Pure Analytic continuation of currents \ beginAppl{ perioda l i g nand Math∗} division period problems 23 open parenthesis, Forum Math 1 9 .70 1 ( closing 1 989 ) parenthesis, 1 5 – 5 1 . comma 145 endash 1 50 period \ tagopen∗{$ square ( bracket 22[3] ) 2$closing C.A.Berenstein,R.Gay,A.VidrasandA.Yger,} P square{ 1 } bracket(D) .. C period\ Aphi period= B e r . e n s. t e i . n comma =Residues P R period{ andn B G + a y and 1 } A (D) e´\zoutphi Identities= 0, Progr . Math . 1 1 3 , Birkh a¨ user , 1 9 93 . period Y g e r comma Analytic continuation of currents and division n \endproblems{ a l i g n comma∗} [4] Forum C.A.BerensteinandB.A.Taylor, Math period 1 open parenthesis 1 989 closingInterpolation parenthesis comma problems 1 in5 endashC with 5 1 period open squareapplications bracket 3 closing to harmonic square analysis bracket .., J C . Analyseperiod A Math period . 37 B ( e 1r 980 e n ) s, 1 t 88 e i – n 254 comma . R period G a y comma A\noindent period V i dcan r a[5] s be and represented− A − period − , − Y− −g e, in rOn comma a the unique geometry Residues way of andinterpolating in B the acute-e form varieties zout of Identities, a in series : S commae´ minaire of exponential Lelong – solutions ofProgr the period systemSkoda Math ( 1 period 22980 – ) 8 1 , 1 1 , namely 3 Springer comma , Birkh Heidelberg a-dieresis , 1 – 2 user 5 . comma 1 9 93 period [ 6 ] C . A . B e r e n s t e i n , B . A . T ay l o r et A . Yge r , Sur les syste` mes d 0 open square bracket 4 closing square bracket .. C period A period B e r e n s t e i n and B period A periode´ T ay l o r comma\ [ \ begin Interpolation{ a l iquations g n e d problems}\ diffphie´ rence in C( to- the diff t powere´ ,rentielles of x n with ), Ann applications = . Inst\sum . Fourier to c ( Grenoble{\zeta ) 33} ( 1 983e ˆ ){ , 1i 9 – 1 ( t \tau +harmonic x \cdot analysis30 . \ commaxi J) period} . Analyse\\ Math period 37 open parenthesis 1 980 closing parenthesis comma 1 88 endash\zeta 254 period\ in [ 7V ]\end C . A{ a . l B i g e n r e ed n}\ s] t e i n and A . Yger , Ideals generated by exponential polynomials , open squareAdv bracket . 5 in closing Math square . 60 ( 1bracket 9 86 ) ,emdash 1 – 80 . comma emdash comma On the geometry of interpolating varieties comma in : S acute-e[8] minaire− − − Lelong, − − − endash, On L Skodaojasiewicz 1 980type endash inequalities 8 1 comma for exponential polynomials , J . Math \noindentSpringer commaThis. Anal Heidelberg s . Appleries . 1 commais 29 convergent( 1 9 1 88 endash ) , 1 66 2in5 – 1period 95 the . topology of $ E ( R ˆ{ n + 1 } ) ($open resp square $. bracket , 6 closing Dˆ{\ squareprime bracket} ..( C period R ˆ{ An period + B e 1 r} e n s) t e i ) n comma . $ B period A period T ay l o r et A period Yge r comma Sur les syst to the power of grave-e mes d quoteright sub e-acute quations diff acute-e rence hyphen\ centerline { References } diff acute-e rentielles comma Ann period Inst period Fourier open parenthesis Grenoble closing parenthesis 33 open parenthesis[ 1 ] \quad 1 983M closing . F . parenthesis A t i y comma ah , 1 Resolution 9 endash 1 30 of period singularities and division of distributions , Comm . \quad Pure Applopen . square Math bracket . 23 7 ( closing 1 9 70 square ) , bracket 145 −− .. C1 period 50 . A period B e r e n s t e i n and A period Yger comma Ideals generated by exponential polynomials comma Adv period .. in [ 2Math ] \quad periodC 60 .open A .parenthesis B e r e 1 n 9 86 s closingt e i parenthesis n , R . Ga comma y 1andA endash . 80 Yg period e r , Analytic continuation of currents and division problemsopen square , Forum bracket Math 8 closing . 1 square ( 1 989bracket ) emdash, 1 5 −− comma5 1 emdash . comma On L-suppress sub ojasiewicz type inequalities for exponential polynomials comma J period Math period Anal period Appl period [ 31 29 ] \ openquad parenthesisC.A.Berens 1 9 88 closing parenthesis te in comma ,R.Gay 1 66 endash ,A.VidrasandA.Yger 1 95 period , ResiduesandB $ \acute{e} $ zout Identities , Progr . Math . 1 1 3 , Birkh $ \ddot{a} $ user ,1993 .

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Y g e r , Analytic B e´ zout identities , Adv . in Effective\ centerline Une formule{( 1 989 B acute-e ) , 5zoutde 1 −− estimates74 . } Jacobi sub et in ses Q open square bracket cons e-acute quences sub Appl . Math . 1 0 comma to the power of z sub 1 comma period period period comma z sub n closing square bracket comma Acta sub Ann ( 1 989 ) , 5 1 – 74 . sub\ [period [ ˆ{ [ to} theˆ{ power1 0 of Math ] } sub{ 1 Sci period 1 ] to} the −−− powerˆ{−−−{ of period, 66}} sub{ , E-acute} −−− sub cole{ −−− to the }ˆ{ power, } of{ open, } Eparenthesis f f e c t i 1 v e9 9 1{ subUne Norm formule to the power} B of closing\acute parenthesis{e }{ zout comma{ de sub}} periode s t i m toa t ethe s { powerJacobi of 69} endash{ et 1} 20in sub { s e s } Q[ { cons \acute{e} quences }ˆ{ z { 1 } , . . . , z { n } ], Sup period to the[10] power of−−− period, 24, z1,...,zn],Acta Math. (1991), 69−−120. [ −−−, −−−−−−,EffectiveUneformuleBezoutdeestimatesJacobi´ etinsesQ[consequences´ ,Ann .Sci.66 ´ coleNorm .Sup. 24 Actaopen} parenthesis{ , } 11]{ Ann 1 99 1}ˆ closing{ Math parenthesis} { . } commaˆ{ . 363} { endashS c i 377 . period} 66 {\acute{E}}ˆ{ ( 1 9 9 E 1 } { c o l e }ˆ{ ), } { Norm }ˆ{ 69 −− 1 20 } { . }ˆ{ . } { Sup . } 24 \ ] open square( bracket 1 99 1 ) 1 , 2 363 closing – 377 square . [12] bracket− − −emdash, − − −comma, A bout emdash Ehrenpreis comma ’ Fundamental A bout Ehrenpreis Principle quoteright, in Funda- mental Principle: comma Geometric in : .. and Geometric Algebraic and Aspects Algebraic in Several Aspects Complex Variables , C . A . Berenstein and D . in Several Complex Variables comma C period A period Berenstein and D period C period Struppa open parenthesis \noindent (C 1 .Struppa 99 1 ) ( eds , 363 . ) ,−− Editel377 , Rende . , 1 99 1 , 47 – 61 . [13] − − − , − − − , Formules de eds period closingrepr parenthesise´ sentation comma int e´ grale Editel et comma probl Rendee` mes de comma division , in : Diophantine Approximations $1 99 [ 1 comma 1 2 47 endash ] −−−{ 61 period, } −−−{ , }$ A bout Ehrenpreis ’ Fundamental Principle , in : \quad Geometric and Algebraic Aspects in Severaland Complex Transcendental Variables Numbers , , C P . Philippon A . Berenstein ( ed . ) , Walter and de D Gruyter . C . , Struppa Berlin , 1 9 ( 92 eds , 1 5 . – ) , Editel , Rende , open square37 bracket . [14] 1 3− closing − − , − square − − ,D bracket− modules emdash and exponential comma emdash polynomials comma, Compositio Formulesde Math repr . ( to 1 9 the power of acute-e1 99 sentation 1 , 47 −− int e-acute61 . grale et probl grave-e mes de division comma in : .. Diophantine $ [ 195 3 ) , to ] appear−−−{ . [, 1} 5 ]−−−{ B ., B e}$ r n d Formules t s s o n and de M . A$ n repr d e r s ˆ s{\ o nacute , Henkin{e}} –$ Ramirez sentation int Approximationsformulas and with Transcendental weight factors Numbers, Ann . comma Inst . Fourier P period ( Grenoble Philippon ) 32 open ( 1 982 parenthesis ) , 9 1 – 1 ed 1 period0 . [ 1 closing 6 paren- thesis$ \acute comma{e} Walter$ grale de Gruyter et probl comma $ \grave{e} $ mes de division , in : \quad Diophantine Approximations] I.N.Bernstein, and TranscendentalThe Numbers analytic continuation , P . Philippon of generalized ( functionsed . ) with , Walter respect to de a Gruyter , Berlin commaparameter 1 9 92 comma, Functional 1 5 endash Anal . 37Appl period . 6 ( 1 972 ) , 273 – 285 . [ 1 7 ] J . - E . B j o¨ r k , Rings of Berlinopen square , 1 bracket9 92 , 14 1 closing 5 −− square37 . bracket emdash comma emdash comma D hyphen modules and exponential $ [ 14Differential ] −−−{ Operators, } −−−{, North, } - HollandD − , Amsterdam$ modules , 1 979 and . [exponential 1 8 ] A . B o r polynomials e l , Operations , Compositio Math . ( 1 9 95 ) , to appear . polynomials commaon algebraic CompositioD− Mathmodules period, in open : Algebraic parenthesisD− modules 1 9 95 closing , A . Borel parenthesis ( ed . ) comma , Academic to appear Press , period [open 1 5 square ] \quad bracketB.BerndtssonandM.Andersson, 1 5 closing square bracket .. B period B e r n d t s s o n and M period\quad A nHenkin d e r s s−− o n commaRamirez formulas with weight factors , Ann . InstBoston . Fourier , 1 9 87 ,( 2 Grenoble7 – 269 . [ 1 ) 9 32] ( L . 1 E 982 h r e n) p , r e9 i 1s ,−−Fourier1 1 Analysis 0 . in Several Complex .. Henkin endashVariables Ramirez,formulas Wiley , 1 with 970 . weight [ 20 ] factors D . I comma . G u r e v i c h , Counterexamples to a problem of L . 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[22] S eminaire´ ´ B.T eissier, QuelquesP olytechnique,1972,130. utiles pour la of[ generalized 1 8 ] \quad functionsA . with B o respect r e tol a, OperationsEcole on algebraic $D − $ modules , in : Algebraic $ D − $ modulesr e´ solution ,A des . singularit Borel (e´ s ed,. [ 23 ) ] , B . L i c h t i n , Generalized Dirichlet series and B - parameter commafunctions Functional, Compositio Anal Math period . 65 Appl ( 1 period 988 ) , 6 open parenthesis 1 972 closing parenthesis comma 273 endash 285Academic period Press , Boston , 1 9 87 , 2 7 −− 269 . [ 1 9 ] \quad L . E h r e n p r e i s ,81 Fourier – 1 20 . Analysis in Several Complex Variables , Wiley , 1 970 . open square bracket 1 7 closing square bracket .. J periode´ hyphen E period B j o-dieresis r k commae´ Rings of Differential [ 20 ] \quad[ 24D ] . L I . S . c Gu h w a r r t e z , v iTh c he´ orie , Counterexamplesg n e´ rale des fonctions to a moyenne problem−p ofriodiques L . Schwa, $ r−t { z Operators commaAnn North . of hyphen Math . Holland 48 ( comma 1 947 ) , Amsterdam 857 – 929 . comma[ 25 ] 1 C 979 . S a period b b a h , Proximit e´e´ vanescente , }open$ Functional square bracket Anal 1 8 closing . Appl square . bracket .. A period B o r e l comma Operations on algebraic D hyphen modules 9(1975)II . ,116 Equations−− fonctionnelles1 20 . pour plusieurs fonctions analytiques , Compositio Math . 64 ( 1 987 comma in : Algebraic) , 2 1 3 D – hyphen 241 . modules comma A period Borel open parenthesis ed period closing parenthesis comma [Academic 2 1 ] \quad Press commaL . H Boston $ \ddot comma{o} 1 9$ 87 rm comma a n 2 d 7 eendash r , 269\quad periodThe Analysis of Linear Pa $ t−r $ ialopen Differential square bracket Operators 1 9 closing square , II bracket , Springer .. L period , E h r e n p r e i s comma Fourier Analysis in Several ComplexBerlin Variables , 1 9 83comma . Wiley comma 1 970 period $open [ square 22 bracket ] S 20 ˆ{ closingM }ˆ{ square. } bracket{\acute .. D{e period} minaire I period G}ˆ u{ rL e v i e c h comma j e Counterexamples u n }ˆ{ e to} a{\acute{E} problemc o l e }ˆ of{ Let period} B Schwa . r-t sub T z commae i Functional s s Anal i period e Appl r period , Quelques { Polytechnique } { , 19 open 972 parenthesis , 1 1 97530 closing}ˆ{ c a parenthesis l c u l s } { comma. }$ 1 1 utiles 6 endash pour 1 20 period la r $ \acute{e} $ solution des singularit $ \openacute square{e} bracket$ s , 2 1 closing square bracket .. L period H o-dieresis rm a n d e r comma .. The Analysis of Linear Pa[ t-r 23 ial ] Differential\quad B Operators . L i c comma h t i II n comma , Generalized Springer comma Dirichlet series and B − functions , Compositio Math . 65 ( 1 988 ) , Berlin comma 1 9 83 period \ centerlineopen square{ bracket81 −− 221 closing 20 . } square bracket S to the power of M sub acute-e minaire to the power of period to the power of L e j e u n sub acute-E cole to the power of e to the power of et B period T e i s s i e r comma Quelques Polytechnique\noindent [ sub 24 comma ] \quad 1 972L.Schwartz, comma 1 30 sub period to\ thequad powerTh of $ calculs\acute utiles{e} pour$ la o r r i eacute-e $ g solution ˆ{\acute des {e}}$ singularitn $ \acute e-acute{e} s comma$ rale des fonctions moyenne $ − p ˆ{\acute{e}}$ riodiques , Ann . \quad o f Math . \quad 48 (open 1 947 square ) , bracket 857 −− 23929 closing . square bracket .. B period L i c h t i n comma Generalized Dirichlet series and B hyphen[ 25 functions ] \quad commaC. Sabbah Compositio Math , periodProximit 65 open $ parenthesis\acute{e}\ 1 988 closingacute{ parenthesise} $ vanescente comma II . Equations fonctionnelles pour plusieurs fonctions analytiques81 endash 1 20 , period Compositio Math . 64 ( 1 987 ) , 2 1 3 −− 241 . open square bracket 24 closing square bracket .. L period S c h w a r t z comma .. Th acute-e orie g to the power of e-acute n acute-e rale des fonctions moyenne hyphen p to the power of e-acute riodiques comma Ann period .. of Math period .. 48 open parenthesis 1 947 closing parenthesis comma 857 endash 929 period open square bracket 25 closing square bracket .. C period S a b b a h comma Proximit acute-e e-acute vanescente II period Equations fonctionnelles pour plusieurs fonctions analytiques comma Compositio Math period 64 open parenthesis 1 987 closing parenthesis comma 2 1 3 endash 241 period