Triangular sets for solving polynomial systems : a comparison of four methods Philippe Aubry, Marc Moreno Maza To cite this version: Philippe Aubry, Marc Moreno Maza. Triangular sets for solving polynomial systems : a comparison of four methods. [Research Report] lip6.1997.009, LIP6. 1997. hal-02546252 HAL Id: hal-02546252 https://hal.archives-ouvertes.fr/hal-02546252 Submitted on 17 Apr 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Des ensembles triangulaires p our resoudre les systemes p olynomiaux une comparaison de quatre methodes Philipp e Aubry Marc Moreno Maza LIP Universite Paris Place Jussieu Paris Cedex Tel email aubryp ossoibpfr mp ossoibpfr resume Quatre methodes de resolutionde systemesdequationspolynomiales sont presentees et implantees dans un cadre commun Ces methodes sont celles de Wu Wu Lazard Laz Kalkbrener Kal et Wang Wanb El les sont comparees sur divers exemples avec une attention par ticuliere porteea lecacitela concision et la lisibilitedes sorties Triangular Sets for Solving Polynomial Systems a Comparison of Four Metho ds Philipp e Aubry Marc Moreno Maza LIP Universite Paris Place Jussieu Paris Cedex Tel email aubryp ossoibpfr mp ossoibpfr January Abstract Four methods for solving polynomial systems by means of triangular sets are pre sented and implemented in a unied way These methods are those of Wu Wu Lazard Laz Kalkbrener Kal and Wang Wanb They are compared on various examples with emphasizing on eciency conciseness and legibility of the outputs Introduction In this pap er we are concerned with the following problem given a nite family F of multivariate p olynomials over a eld k and with ordered variables X X X n to describ e the ane variety V F ie the common zeros of F over an algebraic closure of k Such a description is usually given by a nite family fT T g of p olynomial r sets with particular prop erties a link b etween the T and F and an algorithm to i compute the T from F A well developed strategy since Buchbergers work Buc i is the following given an ordering on the monomials to choose for T the Grobner basis of the ideal generated by F and compute it by the Buchbergers algorithm Wu WenTsunin Wu introduced another way of solving algebraic systems which is the one we are concerned with in this pap er In that case each T is a p olynomial set i such that two distinct p olynomials in T have distinct greatest variables Such a T is i i called a triangular set The p oints of V T where no leading co ecient of a p olynomial i in T viewed as univariate in its greatest variable vanishes are called the regular zeros i of T Then in Wus metho d the variety V F is the union of the regular zeros of i the T and this decomp osition can b e computed by the original Wus CHRSTREM i algorithm Wu This metho d has b een investigated in many pap ers Among them Cho CG CG GM Wana Wanb Wus metho d is ecient for geometric 1 problems where the degenerate solutions are not interesting For general problems it seems to b e dicult to obtain an ecient implementation and this metho d may pro duce sup eruous triangular sets Wus algorithm like Buchbergers one dep ends on many choices moreover its result is not uniquely dened A Wus like decomp osition of ane varieties can b e obtained by Daniel Lazards algorithm Laz But in that case the denition of triangular sets has b een strength ened denition in order to guarantee irredundant and more canonical decomp osi tions Our pap er rep orts a rst implementation of this metho d and shows that it can b e ecient In MR M Moreno Maza and R Riob o o rep ort a very ecient implemen tation of another algorithm due to Daniel Lazard Laza and called Lextriangular This last algorithm also computes decomp ositions similar to those of Laz but the input must b e a lexicographical Grobner basis of a zerodimensional ideal Lazards decomp ositions have at least two interesting prop erties On one hand numeric solutions may b e easily obtained from them b ecause Lazards triangular sets are normalized def inition See also section in Laza for more details On the other hand Lazards triangular sets are well suited for describing prime ideals see section in Laz whereas there is no b ound on the minimal number of generators for a lexicographical Grobnerbasis of a prime ideal In Kal Michael Kalkbrener introduced another type of triangular sets called regular chains denition together with another link b etween F and the T In that i case V F is the union of the closures wrt Zarisky top ology of the regular zeros of the T Let us mention an example to see the dierence b etween Wu and Lazards i way of solving and Kalkbreners one We consider the system given by the following p olynomials where the ordered variables are c s c s b a and where the co ecients lie in the eld of rational numbers o n s c s c c s s c a s c c s s b c Our implementation of Lazards algorithm Laz pro duces the decomp osition fT T T g where s b a b s b a b a a T b a a c b s b a s b a b a s b a c b a T a s b c b s b c c b T a b c s s c How this solution has to b e understo o d In T one may arbitrarily choose a and b once ab a and obtain successively the values of the indeterminates s c s c The triangular sets T and T describ e the case a Note that in T one may choose arbitrarily b whereas b in T So where is the case b a It is describ ed by T In fact if we add this equation to the initial system the computed decomp osition is only fT g Now our implementation of Kalkbreners algorithm Kal pro duces the decomp osition fC g where C b a s b a b s b a b a a a c b s b a s b c a s s c b c a s c s b 2 In that case a p oint is a solution of the initial system if it lies in the closure of the regular zeros of the only triangular set ab ove denoted by C Although C and T are dierent their regular zero sets have the same closure which contains the regular zeros of the previous T and T Note that Kalkbreners output is simpler but needs futher computations for the zeros satisfying ab a Triangular sets can also b e used to solve quasialgebraic systems denition In Wanb Dongming Wang prop osed such a metho d by means of Wus triangular sets But Wangs pro cess is dierent from Wus one and seems to b e more ecient In the conclusion of Kal Michael Kalkbrener wrote a comparison with the algo rithms of Ritt Wu and Lazard seems to b e interesting In the conclusion of Wanb Dongming Wang wrote a systematic analysis and comparison among them the elim ination metho ds of Lazard and Kalkbrener b oth theoretically and practically remain interesting for future work The purp ose of this pap er is to compare the metho ds of Wu Wu Lazard Laz Kalkbrener Kal and Wang Wanb In the rst section following Lazb we introduce a coherent terminology to present their sp ec ications Then we study some prop erties of regular chains and their connection with towers of simple extensions denition We also lo ok into the sp ecial case of Lazard sets In the second section we review the sp ecications of each metho d Futhermore we give a recursive adaptation of Wangs metho d which seems to us easier to read than the original iterative description Our implementation of Lazards metho d is based on an algorithm for gcd computations of univariate p olynomials with co ecients in a separable tower of simple extensions The idea is a generalisation of the one of MR This algorithm could not take place here and will b e presented in a future pap er In the third section we discuss exp erimentations on those four metho ds We think that a reasonable comparative implementation should satisfy the following requirements the corresp onding algorithms must b e implemented and run with the same hu man material and software conditions using the same data structures and sub routines to make sure that each computed solution is correct not to only fo cus on timings but also on the legibility of the ouputs and their suitability for further uses A strongly typed and ob jectoriented language is convenient to satisfy the rst require ment ab ove We used the AXIOM computer algebra system JS We dened categories corresp onding to the dierent prop erties of triangular sets packages and domains for the common subroutines Futhermore AXIOM is connected with GB the very p owerful Grobnerengine developed by JC FaugereFau This allowed us the nontrivial Grobnerbasis computations which are needed in order to satisfy the second requirement ab ove Our implementation of each metho d uses the same p olynomial domain constructor Thus a metho d involving particular datastructures like the dynamic sets and dynamic p olynomials Dia could not enter within our exp erimentations However we tested each of our examples with the dynamic evaluation This metho d is only usable for easy examples and cannot compare with the metho ds of Wu