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Mémoires D'étudiants -- Hal-Inria+ Année : 2008

Μελέτη πολυομογενών πολυωνυμικών συστημάτων μέσω πινάκων της απαλοίφουσας

Study of multihomogeneous polynomial systems via resultant matrices

Résumé

Resultants provide conditions for the solvability of polynomial equations and allow reducing polynomial system solving to linear algebra computations. Sparse resultants depend on the Newtown polytopes of the input equations. This polytope is the convex hull of the exponent vectors corresponding to the nonzero monomials of the equations (viewed as lattice points in the Cartesian space of dimension equal to the number of variables). In this work, we consider the case of multihomogeneous systems, previously studied in the case where all equations share the same Newtown polytope. We generalize certain constructions to mixed systems, whose Newton polytopes are scaled copies of one polytope, thus taking a step towards systems with arbitrary supports. First, we specify matrices whose determinant equals the resultant and characterize the systems that admit such formulae. B ́ezout-type determinantal formulae do not exist, but we describe all possible Sylvester-type and hybrid formulae. We establish tight bounds for the corresponding degree vectors, as well as precise domains where these concentrate; the latter are new even for the unmixed case. Second, we make use of multiplication tables and strong duality theory to specify resultant matrices explicitly, in the general case. Our public-domainMapleimplementation includes efficient storage of complexes in memory and construction of resultant matrices.
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Dates et versions

hal-02272319 , version 1 (27-08-2019)

Identifiants

  • HAL Id : hal-02272319 , version 1

Citer

Angelos Mantzaflaris. Study of multihomogeneous polynomial systems via resultant matrices. Numerical Analysis [math.NA]. 2008. ⟨hal-02272319⟩

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