Solving Parametric Polynomial Systems Daniel Lazard, Fabrice Rouillier To cite this version: Daniel Lazard, Fabrice Rouillier. Solving Parametric Polynomial Systems. [Research Report] RR- 5322, INRIA. 2004, pp.23. inria-00070678 HAL Id: inria-00070678 https://hal.inria.fr/inria-00070678 Submitted on 19 May 2006 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. INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE Solving Parametric Polynomial Systems Daniel Lazard — Fabrice Rouillier N° 5322 Octobre 2004 Thème SYM apport de recherche ISRN INRIA/RR--5322--FR+ENG ISSN 0249-6399 Solving Parametric Polynomial Systems Daniel Lazard∗y , Fabrice Rouillier zy Thème SYM — Systèmes symboliques Projet SALSA Rapport de recherche n° 5322 — Octobre 2004 —23 pages Abstract: We present a new algorithm for solving basic parametric constructible or semi-algebraic systems like = n n C x C ; p1(x) = 0; : : : ; ps(x) = 0; f1(x) = 0; : : : ; fl(x) = 0 or = x C ; p1(x) = 0; : : : ; ps(x) = 0; f1(x) > f 2 6 6 g S f 2 0; : : : ; fl(x) > 0 , where pi; fi Q[U; X], U = [U1; : : : ; Ud] is the set of parameters and X = [Xd+1; : : : ; Xn] the set of unknowns. g 2 If ΠU denotes the canonical projection on the parameter’s space, solving or remains to compute sub-manifolds d d −1 C S C (resp. R ) such that (ΠU ( ) ; ΠU ) is an analytic covering of (we say that has the (ΠU ; )-covering U ⊂ U ⊂ U \C −1 U U −1 C property). This guarantees that the cardinal of ΠU ( ) is locally constant on and that ΠU ( ) is a finite u \ C U Ud \ C collection of sheets which are all locally homeomorphic to . In the case where ΠU ( ) is dense in C , all the known algorithms for solving or compute implicitly or explicitlyU a Zariski closed subset WC such that any sub-manifold of d C S C W have the (ΠU ; )-covering property. n C We introduce the discriminant varieties of w.r.t. ΠU which are algebraic sets with the above property (even in the d C cases where ΠU is not dense in C ). We then show that the set of points of ΠU ( ) which do not have any neighborhood C with the (ΠU ; )-covering property is a Zariski closed set and thus the minimal discriminant variety of w.r.t. ΠU and we propose an algorithmC to compute it efficiently. Thus, solving the parametric system (resp. ) then remainsC to describe d d C S C WD (resp. R WD) which can be done using critical points method or partial CAD based strategies. n n d We did not fully study the complexity, but in the case of systems where ΠU ( ) = C , the degree of the minimal discriminant variety as well as the running time of an algorithm able to compute itCare singly exponential in the number of variables according to already known results. Key-words: Computer Algebra, Polynomial Systems, Parametric Systems, Real Roots ∗ [email protected] y SALSA (INRIA) Project and CALFOR (LIP6) TEAM z [email protected] Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30 Résolution des Systèmes Polynomiaux Paramètrés Résumé : Nous présentons un nouvel algorithme pour la résolution des ensembles constructibles ou semi-algébriques n n basiques de la forme = x C ; p1(x) = 0; : : : ; ps(x) = 0; f1(x) = 0; : : : ; fl(x) = 0 ou = x C ; p1(x) = C f 2 6 6 g S f 2 0; : : : ; ps(x) = 0; f1(x) > 0; : : : ; fl(x) > 0 , où pi; fi Q[U; X], U = [U1; : : : ; Ud] est l’ensemble des paramètres et g 2 X = [Xd+1; : : : ; Xn] celui des inconnues. Si ΠU est la projection canonique sur l’espace des paramètres, résoudre ou revient à calculer des sous-variétés d d −1 C S différentiables C (resp. R ) telles que (ΠU ( ) ; ΠU ) soit un revêtement analytique de (on dira alors U ⊂ U ⊂ U \ C −1 U que à la propriété de (ΠU ; )-revêtement). Ceci garantit que le cardinal de ΠU ( ) est localement constant sur U −1 C u \ C U et que Π ( ) est une collection finie de feuillets localement homéomorphes à . Dans le cas où ΠU ( ) est dense U U \ C U C dans Cd, les algorithmes résolvant ou calculent implicitement ou explicitement un fermé de Zariski W tel que toutes C S d les sous variétés différentiables constituant C W ont la propriété de (ΠU ; )-revêtement. n C Nous introduisons la notion de variété discriminante de relativement à ΠU comme étant un ensemble algébrique C d ayant la propriété énoncée ci-dessus dans un cadre général (ne supposant pas que ΠU ( ) est dense dans C ). Nous C montrons que l’ensemble des points de ΠU ( ) n’ayant aucun voisinage vérifiant la propriété de (ΠU ; )-revêtement est C C un fermé de Zariski et par conséquent une variété discriminante minimale de relativement à ΠU et nous proposons un C d algorithme la calculant efficacement. Ainsi, résoudre ( ) revient alors à décrire ΠU ( ) WD (resp. R (ΠU ( ) WD)), ce qui peut être fait en utilisant une méthode de pointsC S critiques ou encore de décompositionC n cylindrique\ algébriqueC n partielle. Nous n’avons pas encore étudié complètement la complexité de l’algorithme que nous proposons, mais dans le cas de d systèmes où ΠU ( ) = C , les résultats connus montrent que l’algorithme est simplement exponentiel en le nombre de variables. C Mots-clés : Calcul Formel, Systèmes polynomiaux, Systèmes paramétrés, Zéros réels Solving Parametric Polynomial Systems 3 1 Introduction In this article, we propose a new method for studying basic constructible (resp. semi-algebraic) sets defined as systems of equations and inequations (resp. inequalities) depending on parameters. The following notations will be used : Notation 1 Let us consider the basic semi-algebraic set n = x R ; p1(x) = 0; : : : ; ps(x) = 0; f1(x) > 0; : : : fs(x) > 0 S f 2 g and the basic constructible set n = x C ; p1(x) = 0; : : : ; ps(x) = 0; f1(x) = 0; : : : fs(x) = 0 C f 2 6 6 g where pi; fj are polynomials with rational coefficients. • [U; X] = [U1; : : : Ud; Xd+1; : : : Xn] is the set of indeterminates or variables, while U = [U1; : : : Ud] is the set of parameters and X = [Xd+1; : : : Xn] the set of unknowns; • = p ; : : : ps is the set of polynomials defining the equations; E f 1 g • = f1; : : : fl is the set of polynomials defining the inequations in the complex case or the inequalities in the rFeal case;f g d • For any u C , φu the specialization map U u; 2 −! n d • ΠU : C C denotes the canonical projection on the parameter’s space (u1; : : : ; ud; xd+1; : : : ; xn) −! −! (u1; : : : ; ud); • Given any ideal I we denote by V(I) Cn the associated (algebraic) variety. If a variety is defined as the zero set ⊂ of polynomials with coefficients in Q we call it a Q-algebraic variety; we extend naturally this notation in order to talk about Q-irreducible components, Q-Zariski closure, .. • for any set Cn, will denote its C-Zariski closure. V ⊂ V d d −1 Solving or remains to compute sub-manifolds C (resp. R ) such that (Π ( ) ; ΠU ) is an C S U ⊂ U ⊂ U U \ C analytic covering of (in that case, we say that has the (ΠU ; )-covering property). This guarantees that the cardinal of −1 U −U1 C ΠU ( ) is locally constant on and that ΠU ( ) is a finite collection of sheets which are all locally homeomorphic u \C U d U \C to . In the case where ΠU ( ) is dense in C , all the known algorithms for solving or compute implicitly or explicitly U C d C S a Zariski closed subset W such that any sub-manifold of C W have the (ΠU ; )-covering property (see some examples below). n C In the first section of the present article, we introduce the discriminant varieties of w.r.t. ΠU which are algebraic sets d C with the above property (even in the cases where ΠU is not dense in C ). We then show that the set of points of ΠU ( ) C which do not have any neighborhood with the (ΠU ; )-covering property is a Zariski closed set and thus the minimal C discriminant variety of w.r.t. ΠU . The Cylindrical AlgebraicC Decomposition [7] adapted to computes a discriminant variety of as soon as the E S F C recursive projection steps w.r.t. Xd+1; : : : ; Xn are done first. In fact, a discriminant variety W is defined by the union of the varieties associated with the polynomials obtained after the n d-th projection step. It is far from being optimal with − respect to the number of connected open sets of ΠU ( ) W , as it contains at least a discriminant variety for any system of equalities and inequalities that may be constructedCwithn the polynomials of . In the case of a partial CAD [6], the induced discriminant variety is smaller but depends anyway on the order in whichE S Fthe projections are done : it is thus not minimal in general.