Matrices in elimination theory
type: Item
instance of: scholarly article
title: Matrices in elimination theory
zbMATH Open document ID: 0943.13005
author: Ioannis Z. Emiris
author: Bernard Mourrain
published in: Journal of Symbolic Computation
publication date: 2000-09-05T00:00:00Z
review text: This is a survey devoted to recent developments of the theory of resultants and corresponding matrices. The theory of sparse elimination, Newton polytopes, mixed volumes, toric resultant is set forth. Matrices for computing the resultants are considered, generalized Sylvester's, Macaulay's, as well as Newton's matrices and, on the other hand, a generalized Bézout construction (in particular, it is proved that any maximal minor of the Bézout matrix is a multiple of the resultant), combining the methods (Dixon matrices). Comparison of the resultant matrices is made from the computational point of view. Finally, applications to the theory of solving arbitrary systems of polynomial equations are described: monomial bases and multiplication maps, system solving by the \(u\)-resultant, by reduction to the eigenvalue problem [see also \textit{D. Cox, J. Little} and \textit{D. O'Shea}, ``Using algebraic geometry'', Graduate Texts in Math. 185 (1998; Zbl 0920.13026) sometimes for more details], degenerate cases. Some examples are used for illustration. A very wide list of references is enclosed.
Mathematics Subject Classification ID: 13B25
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 1369538
zbMATH Keywords: toric resultant
zbMATH Keywords: Bezout matrix
zbMATH Keywords: polynomial equations
MaRDI profile type: Publication
OpenAlex ID: W1968297701
cites work: Q3913059
cites work: Q4864428
cites work: Q4714016
cites work: Residue currents and Bezout identities
cites work: Effective Bezout identities in \({\mathbb{Q}}[z_ 1,\dots ,z_ n]\)
cites work: The number of roots of a system of equations
cites work: Fiber polytopes
cites work: Q4314299
cites work: Computation of a specified root of a polynomial system of equations using eigenvectors
cites work: Q4039749
cites work: Generalised characteristic polynomials
cites work: Improved Algorithms for Sign Determination and Existential Quantifier Elimination
cites work: Q4237376
cites work: On Grau’s Method for Simultaneous Factorization of Polynomials
cites work: A global view of residues in the torus
cites work: Algorithm of polynomial complexity for factoring polynomials and finding the components of varieties in subexponential time
cites work: Q4023642
cites work: Q4324883
cites work: THE GEOMETRY OF TORIC VARIETIES
cites work: The generalized Schur decomposition of an arbitrary pencil A–λB—robust software with error bounds and applications. Part II
cites work: Algorithms for residues and Łojasiewicz exponents
cites work: On the complexity of sparse elimination
cites work: Efficient incremental algorithms for the sparse resultant and the mixed volume
cites work: How to count efficiently all affine roots of a polynomial system
cites work: Q4872788
cites work: Introduction to Toric Varieties. (AM-131)
cites work: Q4293510
cites work: Q4352797
cites work: Factorization of polynomials over a finite field and the solution of systems of algebraic equations
cites work: Solving systems of polynomial inequalities in subexponential time
cites work: Q5689624
cites work: Q3142503
cites work: Q5547252
cites work: Q4002797
cites work: Q5814388
cites work: A Polyhedral Method for Solving Sparse Polynomial Systems
cites work: Bernstein's theorem in affine space
cites work: Le formalisme du résultant. (The formalism of resultant)
cites work: Chow polytopes and general resultants
cites work: Q3935355
cites work: Q4714022
cites work: Q3714169
cites work: Newton polytopes and the Bezout theorem
cites work: Q3743387
cites work: Résolution des systèmes d'équations algébriques
cites work: Q4717951
cites work: A note on testing the resultant
cites work: Q4293528
cites work: Implicit representation of rational parametric surfaces
cites work: A homotopy for solving general polynomial systems that respects m- homogeneous structures
cites work: Isolated points, duality and residues
cites work: Q4336109
cites work: Multidimensional structured matrices and polynomial systems
cites work: Computing the isolated roots by matrix methods
cites work: Q4343586
cites work: Product formulas for resultants and Chow forms
cites work: Q4714026
cites work: On the Worst-Case Arithmetic Complexity of Approximating Zeros of Systems of Polynomials
cites work: On the computational complexity and geometry of the first-order theory of the reals. III: Quantifier elimination
cites work: A convex geometric approach to counting the roots of a polynomial system

Matrices in elimination theory (cites work)
cites work: Toric intersection theory for affine root counting

Matrices in elimination theory (cites work)
cites work: Bounds of traces in complete intersections and degrees in the Nullstellensatz

Matrices in elimination theory (cites work)
cites work: Q4075559

Matrices in elimination theory (cites work)
cites work: Q5287551

Matrices in elimination theory (cites work)
cites work: Q4771357

Matrices in elimination theory (cites work)
cites work: Matrix eigenproblems are at the heart of polynomial system solving

Matrices in elimination theory (cites work)
cites work: On the Newton polytope of the resultant

Matrices in elimination theory (cites work)
cites work: Multigraded resultants of Sylvester type

Matrices in elimination theory (cites work)
cites work: Q5795154

Matrices in elimination theory (cites work)
cites work: Homotopies Exploiting Newton Polytopes for Solving Sparse Polynomial Systems

Matrices in elimination theory (cites work)
cites work: Mixed-volume computation by dynamic lifting applied to polynomial system solving

Matrices in elimination theory (cites work)
cites work: Solving sparse linear equations over finite fields

Matrices in elimination theory (cites work)
cites work: Q4274348

Matrices in elimination theory (DOI)
DOI: 10.1006/JSCO.1998.0266

Matrices in elimination theory (Recommended article)
Recommended article: Q4370163
Similarity Score: 0.8084409
Recommender Run: Recommender Run 2

Matrices in elimination theory (Recommended article)
Recommended article: Q4234291
Similarity Score: 0.8047663
Recommender Run: Recommender Run 2

Matrices in elimination theory (Recommended article)
Recommended article: Techniques for exploiting structure in matrix formulae of the sparse resultant
Similarity Score: 0.79425484
Recommender Run: Recommender Run 2

Matrices in elimination theory (Recommended article)
Recommended article: Hybrid sparse resultant matrices for bivariate systems
Similarity Score: 0.7925064
Recommender Run: Recommender Run 2

Matrices in elimination theory (Recommended article)
Recommended article: Q4418122
Similarity Score: 0.79069686
Recommender Run: Recommender Run 2

Matrices in elimination theory (Recommended article)
Recommended article: Explicit formulas for the multivariate resultant.
Similarity Score: 0.78921366
Recommender Run: Recommender Run 2

Matrices in elimination theory (Recommended article)
Recommended article: Multihomogeneous resultant formulae for systems with scaled support
Similarity Score: 0.7861098
Recommender Run: Recommender Run 2

Matrices in elimination theory (Recommended article)
Recommended article: Entry formula for mixed resultant matrix of bivariate polynomial system
Similarity Score: 0.77936757
Recommender Run: Recommender Run 2

Matrices in elimination theory (Recommended article)
Recommended article: Single-lifting Macaulay-type formulae of generalized unmixed sparse resultants
Similarity Score: 0.7790073
Recommender Run: Recommender Run 2