On Computing Thom Polynomials

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Thom Thom-Boardman the order for formulae new classes; give Thom-Boardman we to to and construction related basic spaces geometric moduli new and some a compactify singularities present we contact series; connection for hypergeometric a formulae discover un- we localization previously particular, some between In methods of polynomials. new calculations Thom both explicit known contribute and we calculate computations, work, explicitly can this of we In where examples polynomials. ago, few years these a 50 only than still more are Thom there by observed was phenomenon above oil hc nydpnso h iglrt n h dimensions, the and the singularity called the polyno- on is poly- a multivariate depends as This only expressed map. which the be nomial, of can classes singularity characteristic the given manifold of source a mial the has in the map locus that a the) known of where well (closure ex- is the It simplest of class degenerate. the cohomology being manifolds; differential analytic) the is (or behaviour ample local smooth of between type maps a of means “singularity” A Thom-Boardman particular classes. in singularity maps, of singularities of polynomials h ujc atro hstei stecmuaino Thom of computation the is thesis this of matter subject The hmpolynomial Thom Abstract ftesnuaiy vntog the though Even singularity. the of CEU eTD Collection hpe .Lclzto fTo oyoil 22 Σ - order Second 4. 17 Chapter vi polynomials viii Thom of Localization 3. Chapter 10 theory singularity on Primer 2. Chapter formula Thom-Porteous The - order First 1. Chapter conventions and Notations Introduction ..Cmiaois61 43 56 46 41 57 40 43 31 30 Σ 28 for 26 coefficients The Σ for coefficients 4.5.2. The 4.5.1. Combinatorics coefficients the for formulae 4.5. Explicit formula 20 17 localization 4.4. The singularities compactifications 4.3.3. Thom-Boardman The 22 for 21 model probe 4.3.2. The 18 4.3.1. Localization formula 4.3. Ronga’s definitions different the 4.2. of Equivalence 16 4.1. 14 13 3.4.3. Σ 3.4.2. Σ viii computations analytic 3.4.1. Some 12 approach algorithmic 3.4. An ix specialization 3.3.1. Principal trick substitution 3.3. The singularities contact for 3.2. Localization 3.1. 10 polynomials Thom Known polynomials 2.3.1. Thom classes 2.3. Thom-Boardman viii 2.2. Singularities 2.1. Gröbner degeneration equations 1.5. Restriction 1.4. localization Equivariant stability resolution and 1.3. embedded polynomial Porteous’ Thom the of 1.2. Existence 1.1. notations various of list A classes characteristic and functions 3. Symmetric 2. Partitions 1. A 2 1 2 , , Σ , III A 11 1 2 , 2 ij al fContents of Table 22 i, 1 iv 40 64 36 32 32 61 CEU eTD Collection ilorpy95 80 Bibliography Appendix hpe .Tidodr- order Third 5. Chapter ..Psfradfrua 88 93 80 82 85 series hypergeometric Basic A.5. formulae Pushforward classes cohomology equivariant of A.4. Localization polynomials symmetric A.3. for Formulae A.2. differentials Multivariate A.1. ..Mrnsnuaiis70 The singularities 5.2.1. Σ Morin for model probe 5.2. The 5.1. A 3 iglrt 74 singularity A 3 ijk 68 68 v CEU eTD Collection sn optr;adta stesbetmte hstei otiue o ihbt new both with to, contributes thesis the even this and hard, matter examples still subject methods. concrete the are new both is and computations formulae that about explicit and However more computers; much polynomials. using know Szenes, Thom now Rimányi, A. we behind R. others, structures Pragacz, Némethi, and P. A. author, Kazarian, the Fehér, M. L. Buch, Bérczi, A. still we course, (when of seek coefficients; we formulae. Furthermore, these (closed) compute combinatorics. prefer to rich are methods very coefficients have effective computationally they the that possible) compact, evidence relatively some unique, is elegant, there motivation is the it alphabet; that difference tive is the form in polynomials this are Schur cases for of small combinations linear even as (eg. mials write cases to concrete easy is in computers). it evaluate home second, to other; today’s hard each localization by very two to intractable formulae; convert are latter to which residue the hard formulae very iterated of down often variations formulae; are polyno- which functions; pushforward roots; Schur etc.) alphabet; Chern formulae; of in difference combination polynomials the linear (eg. for classes; forms poly- Chern different these many in First, in mials means: expressed ‘computing’ what be of can question nomials philosophical the into bumps be quickly can called is locus polynomial the this of map; class of the singularity. details cohomology of the fine classes the of the characteristic polynomial quantitative: on the Thom not made of the and of polynomial be map, type a can the as a This type of expressed is, given type the (that itself. homotopy has singularity the map map a on a the generalization where only given a locus depends that are the behaviour is they of of observation properties sense, (co)homological basic a the in The René behaviour), Thom principle; by local example. this proposed latter of polynomials, manifestation the Thom a of manifold. also the are on 1950, the field around the from vector from manifold a characteristic a a of Euler is of the singularities are: expresses characteristic There phenomenon which Poincaré-Hopf Euler this theorem, the the of map. and examples expresses curvature; well-known the which Two spaces of theorem, (closed) data. 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[1,0,44,-17,1501,-989,47193,-39983,1431989,-1392409,42670891, productSeries 0 repeat : [a] 1 => = a unit Num :: unit unit [a] convolveWith) -> (flip [[(a,Int)]] foldl => = a productSeries Num :: productSeries okr(cefep:et s= xs series ((coeff,exp):rest) = worker _ ys [] terms worker worker = ys 2,),(3,),(1,)]] ] (-11,9) , (-37,7) , (29,2) [ ] ] , (14,5) (-17,3) [ , , (15,2) [ [ coefficient,exponent iWt (+) zipWith rpiaeep0+ a *of)xs) (*coeff) xs) map rest ++ (worker 0 exp (replicate F ( q = ) (1 − q 3.4. a ecmue ihtefnto call function the with computed be can 0 = 15 ar.Tescn rueti nifiiels,rpeetn a representing list, infinite an is argument second The pairs. ) q 2 oeaayi computations analytic Some 17 + q 3 ) · (1 − 14 q 1 5 ) · (1 − 29 q F 2 ejs aet replace to have just we , 37 + q 7 11 + q 9 ) . unit 31 CEU eTD Collection oaltefie onsaeo h aetp.Tefc sthat is fact The type. same the of are α points fixed the all so fcdmnin2 u only but 2, codimension of ar r ( are pairs 1: codimension of algebra quotient of type single a also is There case. for possibility spaces moduli the of geometry the understanding principles, tti om ti la htw a aetelimit the take can we that clear is it form, this At µ r h ae nti case, this In same. the are complicated more into dive we before have introduction we an as Since serves computations. it case; possible simplest the (Σ these of few a rederive will We singularities. these for polynomials Thom classes A.5. Appendix in summarized are use we [ to 32 ytelmtcs .. fthe of A.5.6 case limit the by i : 4 = = al 2. Table GR90 h eesr nu aafrtelclzto oml,ta s h ita agn Euler tangent virtual the is, that formula, localization the for data input necessary The 3.4.1. 3.4.2. q i III sing. − P I A A 1 a,b Q a,b hoe .. ihLma312gives 3.1.2 Lemma with 3.2.1 Theorem ) d 4 r optdi [ in computed are , o h akrudo hster.Tenttosadtefnaetlresults fundamental the and notations The theory. this on background the for ] ,a a, , Σ Σ d I 1 2 0 ( 2 n , , al fsnuaiiso ormntp Σ type Boardman of singularities of Table for ) ( , ν 3 ) y x xy, ( d ± A III , y x xy, = = = 0 namely, , III ( 1 ideal x lim = 2 . n n n X X X i i i , a =0 =0 =0 d 2 − − − 2 a A +1 n . , a 1 1 1 4 ∈ + →∞ y , 1 , gi,tesingularity the Again, ( ) q Q q N III soe nΣ in open is y b ( ; ) b 2 i l i ( htis, that ; n X q =0 ) − i ) − 3 =0 − ) ν Q 1 , Q i 3 1) 1 ± ( ( FR08 µ Σ , a a q q l i ( =0 = − q i | l 0.S h hm“eis ossso igetr nthis in term single a of consists “series” Thom the So (0). = ; q + + q ν bnma theorem. -binomial 2, = 1 − q ; ( + 2 ( (1 J s ) q µ i d , b b − +1 1 | n ( 2 q ) n y“ees niern” sn ale opttosof computations earlier using engineering”, “reverse by ) (2) − − − i i = − 1) o ml iglrte ( singularities small for ] ν ) − ( ) i q ± 1 − codim 2 1 1) · i q | / ; hsterTo oyoil r h ae hsis This same. the are polynomials Thom their thus , q 1 ν Q i q ( ( ( = ( − − − x ) i 4 2 l n +1 l n | 2 2 = ) q − − − − y y xy, , and i a, ) i · 1 +1 i ) ofs ( 2( = − ( 0 a a − − − 1 ( − − III 1) q 1) a ` = l 2 .Teeaetotpso utetalgebras quotient of types two are There ). 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Here 58 Proposition a resp. with i and tflosdrcl rmti ento htΣ that definition this from directly follows It ecnepottesmlct fteecssbcuethe because cases these of simplicity the exploit can We ewl s h olwn elkonpoete ftemap the of properties well-known following the use will We (iii) (ii) curry (i) t GL n htis, that , b en omlvral.(hscrepnst let to corresponds (This variable. formal a being case. j m atrzto formula factorization with ups that Suppose λ im ker r hnteeuvratCencassof classes Chern equivariant the then are ; n ersnain,vee as viewed representations, ntefloig ewl ihrasm that assume either will we following, the In eoe h rsrcino h)ntrlisomorphism natural the) of (restriction the denotes ( ( % = % n,m n,m ` 4.4.1 i ( : corank β ) ) ) n < m ssandb h mgso h cu polynomials Schur the of images the by spanned is ssand(over spanned is % ≤ . n,m Σ curry Σ ( n m ϕ ij m ij A : rmteTo eispito iwi seog ocnie the consider to enough is it view of point series Thom the from , ( % ( : and n Z A ⊂ % A A,B C B , [ n,m : i i c B , n λ B , Hom 1 m c , X k ` → : ( but ∞ m =0 ( ]cnb bandb h specialization the by obtained be can )] m s Z 2 α e λ − c , − [ ( C = ) ( c % n A n ( Tp c 1 3 m n,m + c n + )= )) Z c , . . . , i ⊗ i = ) i ( yteShrpolynomials Schur the by ) 1) + = ) ) r 2 GL ( α ,C B, c , ) c ] 1 = i s ∈ n M n 3 ) { → ( ( . . . , t ≤ m n 0 × e Z i − ,s e swr ihtefra eso instead: version formal the with work us let so ), m ) × } ( +1) = [ Z Hom GL ) c c −−→ rank n ( [ 1 i ] ∼ a b c , ij 7→ → + 1 Σ m 6⊂ + 1 1 seFgr ) hnw aet ocalled so to have we Then 8). Figure (see − a , ( 2 • ( euvratvco ude vrtepoint; the over bundles vector -equivariant ( Hom ,m n, c , λ ,j ,B A, H c ϕ a 2 B i G htis, that ; P ( ) P A ,ee if even ), 3 a . . . , G ∗ ,B A, ( s . . . , B ( − j m α i n ( sepyif empty is ) resp. A =1 =1 )) A, m ( − − A ) ) ] · n a b ⊗ ≥ Hom a hti,teeeit universal a exists there is, that , A ; j J ⊂ i ) Σ t t b B H ) stability n j i A s 1 λ • b , β rdefine or , n < m ). ,j G ∗ ( 2 ( a h form the has and ( ,C B, ( 2 ( b ,B A, % B ,B A, s b , . . . , ) A n,m . λ ) i < n ( and c s B )) h hmpolynomial Thom The : oeta huhthis though that Note . . ) ) λ ); . ( with . m etestandard the be % c ) B A,B n ti particularly is it and ] corank with are ( Tp n λ +1) ( ( ( = G m n sthe as -equivariant +1) − n ( m n m +1) ,where ), ( m + source +1) ,α β, ⊂ GL λ 6⊂ n ) ; CEU eTD Collection i obntra.Teeaetoseilcsswihaesmwa air namely, easier, somewhat are which cases special two are There combinatorial. where Theorem 4.2.2. Theorem in proved we as coefficients the about anything h rtcs atal ti nuht consider to enough is it i (actually case first the (15) in otecoefficient the to sign) [ eg. in found be can proof A fShrplnmas(ep cu lse,t eprecise) be to classes, Schur (resp. namely, polynomials 1.4), Schur Section of (see equations restriction as Σ interpreted restricting are also we be can equations These s ( ( ;i hs ae,w ilgv lsdfrua o hs offiins nte interesting Another coefficients. these for formulae closed give will we cases, these in 1; = r n + m Computing osmi p epoe h olwn theorem. following the proved we up, it sum To rtn h nvra polynomial universal the Writing above: observations the to result this Compare i ) ) ( ⊂ b e − αβ λ a n h eodcs ( case second the and ; ∈ sjs h euvrat ue class Euler (equivariant) the just is ) 4.4.2 iue8. Figure Z r endb 1)a h cu offiinso h class the of coefficients Schur the as (15) by defined are e . αβ n h hmplnma fthe of polynomial Thom The em ob rtydffiuti eea;hwvr h iclisaemostly are difficulties the however, general; in difficult pretty be to seems ij % k,r oΣ to e + αβ e h hp of shape The k Tp k Tp of β ( for r ( [Σ s ( r = ) α n e FP98 = ) • ( m k d ,j a λ k ) ≤ ) ( Tp s ,b a, X hr ( where α,β = β i ( s 0 ( . ,Scin32 oethat Note 3.2. Section ], Tp r b ( ]= )] λ i ( n = ) ntecas[Σ class the in ) en hti ( if that means ) − m ( pern ntefcoiainformula. factorization the in appearing r 1) ) m n h ls [Σ class the and ) ( X X i b α,β | α λ 1) + − | e e d αβ a α Σ αβ λ k ) ( e ij · · r · = ( + · s s [Σ Hom iglrt nrltv codimension relative in singularity s λ ( i i +1) i α ( • ( − r c ,j ( + ) • a ( ( i ,j )masta if that means 1) ) ) ,b a, ⊂ ,B A, i + s ( 1) + ,r i, β β, λ ( )] α • e b otntl,teeaealzero, all are these fortunately, ; ,j )(e pedxA.2). Appendix (see )) ) ) + ( , ( ( c r ,b a, i ) s + ] oeta eddntsay not did we that Note )]. , α k i < k i +1 ( ]a iercombination linear a as )] − = ) a i. 6⊂ ( = ) ; [Σ λ d , λ • ,j − d snneo then nonzero, is ( λ 1) ,b a, qas(pto (up equals r | α )] = | s . α e − ( a i ,and ), or 1 + r is 59 CEU eTD Collection is bev htteiaeo h map the of image the that Observe section a us gives ϕ with (compare Proof naturally. vrafixed a over where (16) oee,tepolmo expanding of problem the However, oto hc susle.Teeaesm rcal pca ae,though: cases, special tractable some problems, expansion are similar There of family unsolved. larger is a which to of belongs question most This general. in unsolved is over j 1) o hs ehv osprt h “variable” the separate to have we this, For (16). thus section, zero the to transversal is We unsolved. is them compute Theorem to recipe enumerative case an the or compute formulae can giving of problem the is case 60 u fcourse, of but pae in -planes s egtasection a get we ’s, Σ aual,oewudtyt pl h uhowr oml TermA41 oevaluate to A.4.1) (Theorem formula pushforward the apply to try would one Naturally, • (b) ,j (a) Gr n tsoet-n nΣ on one-to-one it’s and , e j j suul steeuvratElrcas ihtegroup the with class, Euler equivariant the is usual, as , ( j acu n[ in Lascoux r top A = Hom + Hom :i hscase, this in 1: = ) 4.4.3 Then . A J i hnw a ieanc obntra nepeaino h offiins but coefficients, the of interpretation combinatorial nice a give can we when , i hr ls,afruai rvdi [ in proved is formula a class, Chern ∈ :i hscase, this in 1: = oteoepitsae and space, one-point the to ker ( π ( A . Gr ∧ ∗ y   ( ⊗ e Let 2 σ q j ( ,B A, ( J ,B A, ( Hom ϕ A ) [ Las78 σ pr π ftevco bundle vector the of ) LP00 = ∼ .Nt that Note ). i = ) ftebundle the of 2 : = = ) Σ ( | Hom Gr A Z • j ,j :

j ker LmaA.2.8); (Lemma ] ] ,however. 2, = ( ( eto 3). Section , Z A A ( s ,B J, A A λ ( i = i A B , q ( ) • ∧

λ A J

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− A Hom J + → [Σ y   A 1 pt pr ∨ i s J q ) = λ ∨ • ⊗ 2 λ A Gr 0 = ) ). ( ,j k eoetepoeto a rmteGasana of Grassmannian the from map projection the denote Z ⊗ ⊂ ( A = B etitdt h eolocus zero the to restricted sspecial; is ) = ] ⊂ A Sym j ( B r ( ssot) rmta,i olw that follows it that, From smooth). is X osdrtediagram the Consider

+ π A X ϕ,ψ ker ⊗ J ∗ i π ker ) ) J j e 2 ( →∧ −→ J −→ ∗ i × ( ( ,B A, into )

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) ) · Z J J (( ( 1) s → → = ∼ q h rtse is step ﬁrst the ; s ∨ ( ϕ π sarslto of resolution a is J (1 Sym | 2 ( ⊗ λ ker )); ∗ Hom Gr A J k | A ]; y   s A ) ∨ ) { j ) q ( s sjs the just is ∨ λ 2 e ⊗ ( q J ψ

,B A, ( A ⊗ A ( B B ( Sym ;tu any thus ); .Cmiigteefrdiﬀerent for these Combining ). J ,π J, B ) )(17) · ) π . 2 ∗ −→ ,B A, ∗ . B s λ ) Z k ( , f GL A hCencas o the for class; Chern th ) ϕψ ftesection the of Σ J

ϕ ( • ∨ A ,j ⊗ eecmue by computed were ∈ J ) i scerthat clear is (it J ) × Hom . ∨ ⊗ GL B ( ( Sym B ) σ 2 acting ,B A, σ ) CEU eTD Collection Lemma A.2.8: otecdmnino Σ of codimension the to Theorem nescinΣ Σ since intersection formula, Giambelli-Thom-Porteous the of situation [ in noted as Proof. [ in found be can twisting of explanation general A simply the becomes by corank’ ‘curried the then and matrices of root Chern L only the formally: understood Remark. Theorem line’ base ‘shifted of coefficient us let so of important, Theorem coefficients very this the theorem: find that We a shows as It combinatorics. it rich motivation: very repeat a have as polynomials serves Thom (c) the Case approaches. different with Proof. n hnteCenrosof roots Chern the then and , nti eto ewl elwt h obntrc fteseilcssmnindabove. mentioned cases special the of combinatorics the with deal will we section this In et e ssaetoterm bu h ae a n (b). and (a) cases the about theorems two state us let Next, them handle but (b), and (a) cases for directly 4.4.3 Theorem use not will we Nonetheless, ...Tececet for coefficients The 4.5.1. (c) h oieso fΣ of codimension The of elements the that Notice wse ymti eeeaylocus degeneracy symmetric twisted ([ stiil oteei opsfrad hstececet nteTo eisare series Thom the in coefficients the coefficients thus the pushforward; as no same the is exactly there so trivial, is i mletnnrva case nontrivial smallest Las78 = oeta h iebundle line the that Note 4.4.4 4.4.5 4.4.6 LP00 s j 1 ν nti case, this in : ( + . ,A A, . . Σ ( ]) A h hmsre fthe of series Thom The [Σ [Σ • ( . ,weeasmlrdgnrc ou rbe scniee—eaei the in are considered—we is problem locus degeneracy similar a where ], , m i 1 eoeby Denote ) • • ∨ ( , ,j A 1 − nteepninof expansion the in ⊗ ( ( n A A n B , W i 1 i + L , B , ( E ,A A, ) m ,n i, 1 ∩ λ/µ r ) ]=2 = )] + • A Hom , i E [Σ = 1 ∨ ]= )] ( − A ( n λ/µ

A ⊗ ∨ = ) i i 4.5. ) n j ( J Σ ⊗ f iue3). Figure cf. , ( B Hom Sym = 1 · B , c n s i, ( i ) L = ( √ det ,A A, b ) 1 r j j m . e Combinatorics + ⊂ h determinant the L a osur ot otefruaaoesol be should above formula the so root, square no has 2 ( ( Sym ) i ewl edtefloigsaeet fLemma of statements following the need will We A ,B A, ,seScin452 oeta nti case this in that Note 4.5.2. Section see 2, = s Sym − Σ are Hom ν ∨ ⊂ 1)+1 ∨ − ii ⊗ µ oml ([ formula λ ⊗ ( 2 f 2 Sym Hom iglrt is singularity j A ): i − √ ϕψ C W ( ( √ + + hsi novd u ecncmuethe compute can we but unsolved, is this ; A α ,A A, L i ! L , ]= )] 1 ∨ n n 2 C ( ) is A + Sym ⊗ − − FNR05 a eietﬁdwt symmetric with identiﬁed be can ) where ∨ i β/ β/ ) B j i c Nt hti hscatrw s the use we chapter this in that (Note . ⊗ corank HT84 mn 2 where 2 2 − ,B A, B , . . . , − i,j A = ) b • n Ts ∈ , j +1 ]. ) 1 e otecasi usini given is question in class the so , n . is ) ,[ ], ( × − A ( = = ( A n Hom n α JLP82 β P . ∨ B , n mn ,j j, = ⊗ + f m ( ν A β B − β/ ± − sjs h transversal the just is ) 1 rs ⊗ 1 ,[ ], − 2. n steCenro of root Chern the is , . . . , ν ± ,B A, A ,wihequals which 1, + Pra90 where , ) . 2 ;so—exactly ); , 1) ,[ ], . f ν Ful96 ± sthe is i × ]). 61 π i CEU eTD Collection h aeo ovnec,w aclt h oa hr class Chern total the calculate we convenience, of sake the formula Proof. and where where 1 substitute we where ` Remark. 62 h hmsre agae hstermcluae h lws ere ato h Thom the Theorem of part degree” “lowest the calculates theorem this language, series. series Thom the Remark. Σ Theorem determinant if the of polynomial expansion the the in is that observes (one [ computation ward eg. see literature, polynomial polynomial symmetric Schur skew the) of expansion ( ij λ µ ( ) K iial,Term446lasto leads 4.4.6 Theorem Similarly, following the implies immediately 4.4.5 Theorem with together lemma, The − ` , 6⊂ µ (2) (1) i = ( x b λ nteRS;then RHS); the in µ codn oTerm442 oepes[Σ express to 4.4.2, Theorem to According 1) + j . e i hsi o h ae n hudtk ( take should one case, the not is this (if ) c usoe the over runs Let utemr,if Furthermore, Then respectively. bundle, ih ste‘tiwy partition ‘stairway’ the is ( hsi h mletrltv oieso hr h iglrt per tal In all. at appears singularity the where codimension relative smallest the is This u notation Our ,µ λ, − 4.5.2 4.5.1 nrltv codimension relative in i +1 A : ) n ( m . X A . and ≥ sn h hrhn notation shorthand the Using ∨ Q h hmplnma ftescn re hmBada singularity Thom-Boardman order second the of polynomial Thom The λ 0 E ⊗ c Σ ( ⊂ λ/µ m n /k ij B B Σ ( 0 i + X ( A h ( - m i, o the for ! s − − k 1 n µ , E L ∨ 1 λ i i = ) ( ( A eune flength of sequences λ/µ ean be − c ⊗ r 1) + A saln udeand bundle line a is 1 k ) nolna obnto fpout fShrplnmas For polynomials. Schur of products of combination linear into ) d ( j B ≤ ⊗ ( A s pt clr.A motn oolr sthat is corollary important An scalar). a to up ) = n = λ/µ − L i, smtvtdb h olwn oml.Spoethat Suppose formula. following the by motivated is ) ⊗ k codim ( = = ) n λ,µ A heeetr ymti oyoili h (Jacobi-Trudi the in polynomial symmetric elementary th | B dmninlada and -dimensional X λ µ = ) r b = ) X 1 | ) ⊂b ∈ j µ X , + = e K ⊂ 2! j 1 Σ e   ( = | λ µ s − µ 2 , ij E ⊂ s ( Sta99 X k | | 3! i i µ ( X 1 λ/µ ≥ h λ λ/µ ,j j, = − |− 1 = + ⊂ 0 , i λ,µ m j c 4! ih 1 ( ( ti scalled is (this )= 1) + k ` − n j n ( . . . , ( − ) h is ) h ro ftefruai straightfor- a is formula the of proof The ]). − λ E µ A ) 1 1) · x · ) = λ/µ , . . . , ∈{ ∨ c spriinwith partition is i and , / 1 λ 2 i, 1 + ⊗ X 0 ( r 1 1 · ( , L λ , . . . , 1 l ehv od st xadthe expand to is do to have we all ] · i + n ( E B } ) i,j + 2 , ) ` | b ( m λ ) i s , Y µ j ) ,   |−| and e µ 1) ∈ E ) /µ dmninl(qiain)vector (equivariant) -dimensional j ( λ/µ { +1 A µ · 2 n ( λ/ e and xoeta specialization exponential |   i ) n ( and ) · µ ( ) s ( s n . X µ l { µ · ≥ − − λ e ( s + 0 ( A x ( x B d ) c i e ) −| ` l ) ( savldpartition valid a is ( − ( µ i µ − λ ) 1 | j , ) A µ , . . . , µ e ) ≤ ) , )   E n λ/µ . then , n nta of instead ) ( n E ahterm each ) λ/µ ( n nthe in

0 = ) n . ≥ λ CEU eTD Collection ti nes xriete opoeta nsc situation a such in that prove to then exercise easy an is It that Observe etrdb h uhr(ae nnmrcleiec) n neednl yPaaz and con- Pragacz; was by This independently polynomial: and Thom evidence), any [ numerical for in on proved general, (based then in author true the is by same jectured The polynomials. Schur that fact the that using such directly, sequences follows 0-1 theorem over the runs sum second the where any numbers the that Note xadn h determinant the Expanding Proof. where (18) Theorem ewl get will we sn h em,tePeifrua and formula, Pieri the Lemma, the Using ert sfollows: as rewrite oeta nbt ae,teTo oyoili ongtv iercmiainof combination linear nonnegative a is polynomial Thom the cases, both in that Note ihsm ok ecngtamr lgn oml.Itouetenotations the Introduce formula. elegant more a get can we work, some With β ∈ K codn oTerm452 h coeﬃcient the 4.5.2, Theorem to According C = . c 4.5.3 { ( ( A n k ,µ ν, a ∨ ν,µ . W7,PW07b PW07a, ⊗ := : ) a a so h form the of is B ν,µ ν,µ X j =0 − ν k = = ⊂ A x det = ) n k n ∈{ h j

i c X ` , 0 ( [Σ lofr acllk triangle: Pascal-like a form also , E − 1 µ } ⊂ ( ν/α i, A B X µ ν µ µ λ 1 1 k = ) k,l ) ⊂ ( l 62 132 31 26 16 6 1 det E and ( r + + i ≤ i h ]= )] ν/ ]. n eragn h usyields sums the rearranging and ) = 11 16 15 11 5 1 ( i x i X i, l ( A ∈{ ≥ − µ e − 8 7 4 1 − 0 X − and 0 ( k l x ν,µ c , 4 3 1 if 0 A 1 ) X l B } F e ( k,l l ) − ( ( 2 1 λ/µ ∈ − where ) µ i +1 | = ) K A ) ν ( 1 − = ) ( | − | F n µ 1) ν/µ α E := ) ( l X 1 +1) | µ λ/µ X k µ = µ ( 1 ≥ + ν | µ i a 0 ) k 2 x = det ν,µ if ( ( + | · k α + − E X s 2 n < l l if 0 = ) i i µ = 1) λ/ e ( of i = − µ 2 i − µ h − µ k e + 3 + 1 n s s ( ( λ · · · µ { i } ( k l k det x k ν, i e ) l . ( 1) + h A + µ · + e + savldpriin rmthis From partition. valid a is α µ X ) s ( { ) i µ A ( n =0 n ν, e , i 6⊂ µ µ + e − − + ) E λ x ) k,l k l sasm hc ecan we which sum, a is βB ν/α e and ( ∈ A i ( = ) × k,l i ) ) i s k ∈ { ≥ λ n det ( × B ` n ( ( ) A λ . , ) od for holds ) ` , ( µ ). 63 CEU eTD Collection where ( where (19) sn h oainlcneto that convention notational the using get we row, last the by determinant the Expanding Σ of series Thom where hswa egthr satal lsdfruafrthe for formula closed a Corollary actually is here got we what thus 64 oeta ihsm bs fntto,hr eallow we here Example notation, of abuse some with that Note where x − is,ltu opt h uepesoso h form the of subexpressions the compute us let First, oeta ntefrua(8,tececet r aietyidpnetof independent manifestly are coefficients the (18), formula the in that Note ydefinition, By ...Tececet for coefficients The 4.5.2. y atrvnse rmtednmntr ic h ento of definition the since denominator, the from vanishes factor ) M K stesto quadruples of set the is = 2( 4.5.5 s a ( x a,b,c T 4.5.4 K + − Ts ( − n . 1 b (Σ = ) − K y (2 22 pcaiigfrΣ for Specializing + i,i . ) { 21 2 x, = ) ( r h aea h offiinsi h expansion the in coefficients the as same the are h hmsre fthe of series Thom The c ]= )] ,b ,d c, b, a, s = ubro tnadYugtbeu fsae( shape of tableaux Young standard of number = nti eto,w eieafruafrteenumbers. these for formula a derive we section, this In . 2 ( 2 a,b,c ,x y, b ( + ν n X i c ( + ) K +1 x + (2 ν , + ) s − ,x x, y − 1 a ∈ = ) y Ts +2 : ) ) N a d (Σ n i s + 4 − ( 1 + 1 + n = 2( Σ b,c ` i, 21 y, 0 : ( 1 1 22 x ν ) c , = ) 2 − + 1 n − . − s y = ) ≤ 1 1 = h aaa triangle Catalan the = = ) codn oTerm444 h offiinsi the in coefficients the 4.4.4, Theorem to According 2 y c b ν a = b ) X ± s + 3 1 n ` ≤ ∈ c X ( − Σ

s K +2 = ν r 1 a, i, − ( ( F s 1 x b ,y x, d (2 = ) (2 X ν 1 b i n/ a 0 =0 ( ( +1 iglrt is singularity − + a,b,c M x y 1 + 2 /ν c ≤ ) = ) ) c s y − i, a a + T ( ) r,r +2 c +2 ) a a ( ( +1 n +2 i ≤ ν | ) ) ν − x + s · ,c · − i,i d ,c d, 1 n s ) rs + | − | and s ( ) 1 + ( hmseries Thom 2 + b M s ν ( x b,c y ± (2 (2 ( − n + + ν ) − x a and y r,r ν o hs ealtefc that fact the recall this, For . + + i,i ) ) d − y b b ) b | ) +1 ) +1 ( +3 = b ob addwt zeros. with padded be to · = , ,y x, +1 s rs s a ( i a,b ( 1 c s d,c, + − ) s ( n , ( ) ( n,k : a b 1 x − − +1 (2 (2 = −

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( be will compactification uhthat such that such egtdpoetv spaces projective over weighted bundle a is which H an is there thus action, ω Proof. on action well-defined a induces our Theorem motivated observation this grading; second the action. wrt. homogeneous also are functions n h eodoeis one second the and ieopim.Tealsgvsadw-h-at ento ftecmlxsrcueo the on structure complex curves the test of of definition space down-the-earth a moduli gives atlas The diffemorphisms. ieetgrading different Remark. explicitly: down written also be can functions transition These that such be will Example. u 72 C w i u x v 0 × ∈ i 0 i i i 0 = 0 0 ω ( = egtdpoetv pcsaesnua;hwvr hi iglrte r eyml,namely, mild, very are singularities their however, singular; are spaces projective Weighted This hstermgvsu nalso h ouisaeo etcre:Tecat are charts The curves: test of space moduli the on atlas an us gives theorem This =0 = = = = cso h e direction new the on acts v C i ,ω,ω ωβ, α, ( × ehv opoeta o n two any for that prove to have We ω = − − − x ehave we , 1 x C j w v u · j 3 w j H j × j ω [ oeta hs xrsin for expressions these that Note x 5.2.3 γ x x osdrtetasto ucinfrom function transition the Consider i j 0 j 2 j 5 ) lentvl,tetasto ucin ev h set the leave functions transition the alternatively, ]); +2 · cinalw st mtt h rcs fScin4321 h nta,wrong initial, The 4.3.2.1: Section of process the imitate to us allows action = +5 ◦ ( 2 u Hy ω ,ω γ, · · · v x . j 2 j h rtoei h tnadgrading standard the is one first The : j 7 u The { = j 1 x 0 = ω x 3 = ) i j ω ,... . . δ, − · 1, = C 5 ◦ } deg y u P 1 × H j 2 ω o 1 for ω B n ). H ∼ ( − n ti eyes owiedw:If down: write to easy very is it and , cinon action ihweights with u · x (1) · 1 M i y 0, = ) ( ∈ ω ,u ,w . . . w, v, u, x, 2 = tepoeto a sgvnb [ by given is map projection (the ≤ = d { · M C = Diff v ;1 1; y = w i u x M v i h 2 d ≤ ξ k k 0 k 0 k 0 0 H = u h qiaec relation equivalence the But . J hc a edfie o xml stesto limits of set the as example for defined be can which , C i deg d n d 1 uhthat such (1) d n − = = = = 0 J ◦ ihwih ,ta is, that 1, weight with w M × ( = n h rniinfntosaecmoiin ftwo of compositions are functions transition the and , (1 1 ω d ( ◦ i , u (1 J 2 n , w u x v x 0 = · 1, = ) k k k j n k d y n , ◦ x x ( = ) − x d ) (1 j 2 1 j j 3 / x 1 − − } .I at an fact, In ). − ) Diff ,β . . . β, α, j 3 , . . . , y { − n , 2 x 3 and endby defined 1 u k v ,ω,ω ωu, x, y , k u ) k deg u d / j u eosection zero j 2 (1). j Diff ( x x { d j i ( j 2 J ∈ y x x − hcatt the to chart th − v − 2 j 5 C j 0 d x 2, = ) 2 r ooeeu ihrsetto respect with homogeneous are (1) k N u 1, = = 2 1) v k d ,ω v, / ◦ j deg v n x (1 Z . j Hy j − x k +2 n , j 2 1 3 u ( H +5 deg } } ,... . . w, o oecci group cyclic some for x ω 1 j 0 x uhthat such ) n ewn ofidan find to want we and ; = ) 0 k u = · / u k ( x xss(neednl of (independently exists ξ C j 2 u w x ] v × j 2 j 7 = deg x ,ec h transition The etc. 3, = ) j ) 0 ∈ , H j ∼ = j − ωξ { P hcat h notation The chart. th ( x u ( = u n sdfie yagroup a by defined is w k .Nt htw have we that Note ). − w = ) = j 0 1 j ,β ,δ . . . δ, γ, β, α, x 0 = ,woefiesare fibers whose ), y v j 2 1 +5 deg = } ∼ x and , k ( w v v y j = ) u 2 = n any and , j Z x · · · j { k k · · · − ,then ), x acting 6= 5 i x 0 = 1; = 1, = k ,j i, two C u y H j 3 1 × } ) . 0 CEU eTD Collection suigta gcd( that count assuming to have we (but instead cover finite smooth a degree space natural over projective weighted work a can For we multiplicities): since smooth, are the uemassmerctno rdc) h ako hsmti sclearly is matrix this of rank The product). rk tensor symmetric means ture where swa ewl do. will we what is than less pc ttefie ons ic hs xdpit r yial iglr h egt hudbe should weights the singular, typically are points fixed these Since points. fixed the at space the respect not would compactification the is, that action). torus equivariant; direction’ be ‘extra wouldn’t the pactification on acts torus The (25) space tangent [ nt htwhile that (note es ,snew eoe h eoscin.Nt httern sivratfraltregroup three all for invariant is rank the that Note section). The zero actions: the removed we since 2, least h don)dtrie h map the determines adjoint) the x 1:0:0: 0 : 0 : [1 = ] [ x fcus,w aet ooeieteeutos rte ncnein arxfr,for form, matrix convenient in Written equations. the homogenize to have we course, Of h mg f(h iermprpeetdb)ti arx(reuvlnl,tekre of kernel the equivalently, (or matrix this by) represented map linear (the of image The h egt ftetnetsaeof space tangent the of weights The rmta,w a opt h ou egt ftecrepnigwihe projective weighted corresponding the of weights torus the compute can we that, From d | u aete are they case 4 = | M v | M w i d r h oso h arx h a sntdfie hntern ftemti is matrix the of rank the when defined not is map the matrix; the of rows the are if ] gi,tentrltigt r st lwu h akvreis n nedthat indeed and varieties, rank the up blow to is try to thing natural the Again, . = Diff ξ T Q [ rcl that (recall 0 = x d ] i ξ · · · P 1 cin the action, (1) d d n i sascalar, a is        − 0 rnhdcovering branched ]cnb edo rmTerm522(h rtrwrpeet the represents row first (the 5.2.2 Theorem from off read be can 0] : d , . . . , 1 ξ / ( n/a / ( n/a / ( n/a / ( n/a n h eti h fiber): the is rest the and , 3 . . . M x D π 4 sol sol [ · x N [ 0 D α α α : ohriew hl atrotb h omndivisor). common the by out factor shall we (otherwise 1 = ) [ α : ,u ,... . . v, u, x, 2 2 2 M | 2 x C − − − ,u ,w v, u, x, 1 − x | . . . GL d B : dα 3 2 α ,tu h aki lasa es ;atal,i sat is it actually, 1; least at always is rank the thus 0, 6= ⊂ · · · α α | n A 1 1 1 1 ( ) B ( ) ( ) ( ) ] cin n the and action, t : (1) π ,where 0, = ] x ξ 3 N r etr;mlilcto fvcosi hspic- this in vectors of multiplication vectors; are α α α : ξ ξ M rval ohrieteebdigit h com- the into embedding the (otherwise trivially 99K [ = α [ = ] 2 2 P 3 3 3 C 3 ux x N − − − d − M 3 . . . Gr 2 → → dα 3 2 x 1 α P α α 0 d d [ ∧ 1 d 0 1 1 1 ( P ) P ] J ) ) ) M : [ M n d ihweights with d x ] − ( 2 · · · · · · · · · · · · 1 d C ie yteformula the by given n 1 1 stematrix the is ξ ∧ · · · ∧ )) × ( tatrsfie on of point torus-fixed a at : u · · · action. 2 2 ( ( ( ξx ( 2 + α α α ξux B α : n n n 2 n M x − − − vx N d − . . . d N d dα 3 2 ](24) ) α ] α α , ( = 1 1 1 1 ) ) ) ) d        0 d , . . . , w v u x A d if N ehv a have we ) ξ P ,and 0, 6= n − 1 eg. , 73 CEU eTD Collection Z choosing then and nyasnl lwu ee oee,teei i essmer rsn.W att show to want We present. symmetry less bit a point is any there approaching however, here; that blow-up single a only Proof. 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− 2 j

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,abz/d a, q ,az c, 1 ,b q, − ) ) n

| m z m q ,abz/c q, b/a ,c/b q, . k ( | ( ; b n 2 zq q | ) | ) i < ; m < ( q s b (

1 − ) | zq n 1 = q, z r ) | ; z a d < q /a n ) n 1 , z = qz/a CEU eTD Collection [BV83] B8]Ncl eln n Michéle Vergne, and Berline Nicole [BV82] Brion, Michel [Bri97] Ba7 .McalBoardman, Michael J. [Boa67] Bérczi, László Gergely Fehér, Richárd Rimányi, M. and [BFR02] Be 0]GreyBérczi, Gergely [Bér08] Bz6 egl ´rz n András Bérczi Szenes, and Gergely [BSz06] AG9]V .Anl,V .Vsle,V .Gruo,adO .Lyashko, V. O. and Goryunov, V. V. Vasil’ev, A. V. Arnold, I. V. [AVGL98] G7]PilpGitsadJsp Harris, Joseph and Griffiths Phillip [GH78] Fulton, William [Ful96] [EG98b] Graham, William and Edidin Dan Heckman, J. G. and [EG98a] Duistermaat J. J. Damon, [DH82] James [Dam72] Bott, Raoul and Atiyah F. Michael [AB84] [Ful98] Franz, Matthias [Fra09] [FR07] László Fehér Balázs M. and K˝om˝uves,[FK06] Eisenbud, David [Eis95] Gf3 eec Gaffney, Terence [Gaf83] [FR08] Richárd Rimányi, László Fehér, András Némethi, and M. [FNR05] [FR06] László Richárd Fehér Pragacz, Rimányi, and M. Piotr and [FR04] Fulton William [FP98] nchmlgeéquivariante cohomologie en 225–267. 33 .P rsee,C els .Kney n .Lue,es) otmoayMteais vol. Mathematics, Contemporary eds.), Lauter, K. and Kennedy, G. 2002. AMS, Melles, 324, C. Brasselet, P. J. singularities of theory global 2008. Technology, of University 2006. 93 98 rgnlypbihda o.6i h eis nylpei fMteaia Sciences. Mathematical of Encyclopaedia series: the in 6 Vol. as published Originally 1998, 1993, fterdcdpaespace phase reduced the of 1–28. 1, no. (1984), yps ueMt,vl 0 M,18,p.399–408. pp. 1983, AMS, 40, vol. Math, Pure 241–262. Sympos. pp. 1996, 9, vol. Proc., Conf. Math. Israel Geometry, Algebraic on Conference 65 http://www.math.uwo.ca/ 45–53. pp. 2006, Japan, Soc. Math. 43, vol. Math., Pure in Studies Adv. 1995. Springer, Math. 50 o ,541–549. 4, no. Mathematicae rvdne .. 04 p 69–93. pp. 2004, R.I., Providence, 1998. Springer, Bull. Math. Canad. btutosfrgopactions group for obstructions 595–634. 16) 21–57. (1967), 18) o ,539–549. 2, no. (1983), 120 , , , , , , hmplnma optn taeis survey A strategies. computing polynomial Thom ntesrcueo hmplnmaso singularities of polynomials Thom of structure the On nescintheory Intersection singularities contact of series Thom ´ rsdu hm evcer tcasscharactéristiques équivariantes classes et vecteurs de champ Zéros d’un oaiaini qiain nescinter n h otrsdeformula residue Bott the and theory intersection equivariant in Localization 19) o ,619–636. 3, no. (1998), qiain hwgop o ou actions torus for groups Chow Equivariant 191 hmplnmasfrcnatcassingularities class contact for polynomials Thom cuetvreisi a ude o h lsia groups classical the for bundles flag in varieties Schubert utdgeso iglrte n ordciequotients nonreductive and singularities of Multidegrees h hmplnma of polynomial Thom The ovx-aMpepcaefrcne geometry convex for package Maple a - Convex omttv ler ihave oadagbacgeometry algebraic toward view a with algebra Commutative 20) o ,249–264. 3, no. (2006), 48 iglrte fdffrnibemaps differentiable of Singularities 20) o ,547–560. 4, no. (2005), net Math. Invent. , ∼ pigrVra,19,1998. 1994, Springer-Verlag, , fazcne/ 2009. mfranz/convex/, .R cd c.PrsSe.IMath Sér. I Paris Sci. Acad. R. C. , oisi leri n ocmuaieGoer L McEwan, (L. Geometry Noncommutative and Algebraic in Topics , eladcmlxsnuaiis otm.Mt,vl 5,AMS, 354, vol. Math, Contemp. singularities, complex and Real , Bibliography qiain nescintheory intersection Equivariant hmplnmaso oi singularities Morin of polynomials Thom lse aateitqe euvrats oml elocalisation caractéristiques de équivariantes.Classes Formule rnilso leri geometry algebraic of Principles h oetmpadeuvratcohomology equivariant and map moment The nscn re hmBada singularities Thom-Boardman order second On cuetvreisaddgnrc loci degeneracy and varieties Schubert ntevraini h ooooyo h ypetcform symplectic the of cohomology the in variation the On aclto fTo oyoil n te cohomological other and polynomials Thom of Calculation 69 P 95 1111 ri:8922,2008. arXiv:0809.2925, , 18) o ,259–268. 2, no. (1982), iglrte,Pr Act,Clf,18) Proc. 1981), Calif., (Arcata, 1 Part Singularities, , xrsin o eutnscmn rmthe from coming resultants for Expressions iglrt hoyadIsApplications, Its and Theory Singularity , rnfrainGroups Transformation , nt Hautes Inst. , eeeayo -om n 3-forms and 2-forms of Degeneracy ul odnMt.Soc. Math. London Bull. , hD hss avr,1972. Harvard, thesis, Ph.D. , 295 net Math. Invent. , iglrt hoyI. theory Singularity ie-nesine 1978. Wiley-Interscience, , ulse lcrnclyat electronically published , 18) 539–541. (1982), rc.o h Hirzebruch the of Procs. , hD hss Budapest thesis, Ph.D. , tdsSi ul Math Publ. Sci. Etudes ´ arXiv:math/0608285, , N,vl 1689, vol. LNM, , T,vl 150, vol. GTM, , 131 ueMt.J. Math. Duke , 2 Topology , Fundamenta , 19) o 3, no. (1997), 19) o 3, no. (1998), mr J. Amer. , Springer, , 39 (2007), 23 , CEU eTD Collection Se5 onR Stembridge, R. John Stanley, P. Richard [Ste05] [Sta99] Rossmann, Wulf [Ros89] To6 René Thom, [Tho56] [PW07b] [Mac98] Macdonald, G. Ian [Mac92] Rn2 eieRonga, Felice [Ron72] Richárd Rimányi, [Rim01] Weber, Andrzej and Pragacz Piotr [PW07a] [Pra07] Pragacz, Piotr [Pra88] Sloane, A. J. Neil [OEIS] Mather, N. John [Mat73] [LP09] Pragacz, Piotr and Lascoux Alain [LP00] Lascoux, Alain [Las78] Pragacz, Piotr and Lascoux, Józefiak, Alain Tadeusz [JLP82] Iversen, Birger Tu, W. [Ive72] Loring and Harris Joe Kosinski, A. and [HT84] Haefliger A. Harris, Rahman, Joe Mizan [HK57] and Gasper George [Har95] [GR90] [GHC] 96 P0]SmnPeyton-Jones, Simon [PJ03] Kazarian, E. Maxim [Kaz06] [Por83] Porteous, R. Ian [Por71] [Pra90] http://www.math.lsa.umich.edu/ 313–330. pp. 1989, 173–174, Astérisque, vol. pervers, faisceaux et Orbites représentations, III. http://www.impan.pl/ RIMS; of ah Helv. Math. 499–521. 117–129. (2001), pp. 2007, Math., in Trends moduli, and shtukas, sheaves, 189–199. pp. 1990, 2, 1487–1508. no. 5, Publications, no. Center Banach (2007), 26, vol. ed.), al., et Balcerzyk (S. bra System Compilation Haskell Glasgow The 43–87. Mathematicae Fundamenta http://http://www.research.att.com/ inlProgramming tional 16/0,LM o.12 pigr 91 p 286–307. pp. 1971, Springer, 192, vol. LNM, (1969/70), I toire 385–387. 229–236. (1972), o ,413–454. 3, no. 395–406. pp. 1983, AMS, 40, vol. Math, 233–248. pp. 1973, 1971), Salvador, Bahia, 239–263. pp. 2000, Mathematics, in Trends eds.), symmetries with matrices antisymmetric and symmetric with 23 différentiables 18) o ,71–84. 1, no. (1984), 28 , , , , , , hmplnmasadShrfntos h singularities the functions: Schur and polynomials Thom ylso stoi usae n omlsfrsmercdgnrc loci degeneracy symmetric for formulas and subspaces isotropic of Cycles ymti ucin n alpolynomials Hall and functions Symmetric 19) 5–39. (1992), hmplnmasadShrfntos h singularities the functions: Schur and polynomials Thom hmplnmaso nain oe,Shrfntos n positivity and functions, Schur cones, invariant of polynomials Thom rbn singularities Probing leri emty rtcourse first A geometry. Algebraic 47 e iglrte e plctosdifférentiables applications singularités des Les ´mnieHniCra,vl ,15–97 Exposé 8. 1956–1957, 9, vol. Cartan, Séminaire Henri , eclu e lse ulsaxsnuai´sd ormndodedeux d’ordre Boardman singularités de aux duales classes des calcul Le nmrtv emtyo eeeayloci degeneracy of geometry Enumerative lse eCendu rdi tensoriel produit d’un Chern de Classes xdpitfruafrato ftr nagbacvarieties algebraic on tori of action for formula point fixed A eetPors nItreto hoy(.Elnsu,W utn n .Vistoli, A. and Fulton, W. Ellingsrud, (G. Theory Intersection in Progress Recent , 17) 15–35. (1972), ipesnuaiiso maps of singularities Simple nTo-ormnsingularities Thom-Boardman On hmplnmas ymtisadicdne fsingularities of incidences and symmetries polynomials, Thom qiain utpiiiso ope varieties complex on multiplicities Equivariant cu ucin:Teeadvariations and Theme functions: Schur h nLn nylpdao nee Sequences Integer of Encyclopedia On-Line The nmrtv combinatorics Enumerative eako hmplnmasfor polynomials Thom on Remark 13 h akl 8lnug n irre:Tervsdreport revised The libraries: and language 98 Haskell The al akg o ymti functions symmetric for package Maple A 20) o ,http://www.haskell.org. 1, no. (2003), 195 nsmercadse-ymti eemnna varieties determinantal skew-symmetric and symmetric On iglrte,Pr Act,Clf,18) rc yps Pure Sympos. Proc. 1981), Calif., (Arcata, 2 Part Singularities, , ∼ 20) 85–95. 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