Toric Eigenvalue Methods for Solving Sparse Polynomial Systems

Bender, Matías R.; Telen, Simon

doi:10.48550/arxiv.2006.10654

Cited by 4 publications

(17 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Although the Cox construction is a natural generalization of the familiar geometric quotient P n = (C n+1 \ {0})/(C \ {0}), we believe the application to homotopy continuation is novel. Our work complements the recent use of Cox coordinates for dealing with non-toric solutions in a robust manner in numerical algebraic normal form methods [40,39,7]. Other closely-related work includes the aforementioned polyhedral endgame [24] and the use of toric compactifications in complexity analysis for sparse polynomial system solving [30,31].…”

Section: Introductionmentioning

confidence: 79%

Polyhedral Homotopies in Cox Coordinates

Timothy¹,

Telen²,

Walker³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

We introduce the Cox homotopy algorithm for solving a sparse system of polynomial equations on a compact toric variety X Σ . The algorithm lends its name from a construction, described by Cox, of X Σ as a GIT quotient X Σ = (C k \ Z) G of a quasi-affine variety by the action of a reductive group. Our algorithm tracks paths in the total coordinate space C k of X Σ and can be seen as a homogeneous version of the standard polyhedral homotopy, which works on the dense torus of X Σ . It furthermore generalizes the commonly used path tracking algorithms in (multi)projective spaces in that it tracks a set of homogeneous coordinates contained in the G-orbit corresponding to each solution. The Cox homotopy combines the advantages of polyhedral homotopies and (multi)homogeneous homotopies, tracking only mixed volume many solutions and providing an elegant way to deal with solutions on or near the special divisors of X Σ . In addition, the strategy may help to understand the deficiency of the root count for certain families of systems with respect to the BKK bound.

show abstract

Section: Introductionmentioning

confidence: 79%

Polyhedral Homotopies in Cox Coordinates

Timothy¹,

Telen²,

Walker³

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…The standard method to solve MEP is Atkinson's Delta method [3,Ch. 6,8]. For each 0 ≤ k ≤ α, it considers the overdetermined system F k resulting from F (2) by setting x k = 0.…”

Section: Multiparameter Eigenvalue Problemmentioning

confidence: 99%

“…We refer reader to [27], see also [46], for an algorithm to solve unmixed multilinear systems using Gröbner bases, and to [23,24] using resultants. We also refer to [6,7] for an algorithm based on Gröbner bases to solve square mixed multihomogeneous systems and to [8,48] for a numerical algorithm to solve these systems using eigenvalue computations. However, these generic approaches do not fully exploit the structure of the problem.…”

Section: Multiparameter Eigenvalue Problemmentioning

confidence: 99%

See 1 more Smart Citation

Koszul-type determinantal formulas for families of mixed multilinear systems

Bender,

Faugère,

Mantzaflaris

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula when we can express it as a determinant of a matrix whose elements are the coefficients of the input polynomials. We study the resultant in the context of mixed multilinear polynomial systems, that is multilinear systems with polynomials having different supports, on which determinantal formulas were not known. We construct determinantal formulas for two kind of multilinear systems related to the Multiparameter Eigenvalue Problem (MEP): first, when the polynomials agree in all but one block of variables; second, when the polynomials are bilinear with different supports, related to a bipartite graph. We use the Weyman complex to construct Koszul-type determinantal formulas that generalize Sylvester-type formulas. We can use the matrices associated to these formulas to solve square systems without computing the resultant. The combination of the resultant matrices with the eigenvalue and eigenvector criterion for polynomial systems leads to a new approach for solving MEP.

show abstract

“…This is best understood in the language of toric geometry [24]. Situations in which there are finitely many solutions at infinity (see Assumption 1) can be handled by introducing an extra randomization in the algorithm, which was first used in [41,11]. Where classically the multiplication matrices represent 'multiplication with a polynomial g', the multiplication matrices in these papers represent 'multiplication with a rational function g/ f 0 ', where f 0 is a polynomial that does not vanish at any of the solutions to the system.…”

Section: Introductionmentioning

confidence: 99%

Yet another eigenvalue algorithm for solving polynomial systems

Bender¹,

Telen²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

In latest years, several advancements have been made in symbolic-numerical eigenvalue techniques for solving polynomial systems. In this article, we add to this list by reducing the task to an eigenvalue problem in a considerably faster and simpler way than in previous methods. This results in an algorithm which solves systems with isolated solutions reliably and efficiently, outperforming homotopy methods in overdetermined cases. We provide an implementation in the proof-ofconcept Julia package EigenvalueSolver.jl.

show abstract

Toric Eigenvalue Methods for Solving Sparse Polynomial Systems

Cited by 4 publications

References 25 publications

Polyhedral Homotopies in Cox Coordinates

Polyhedral Homotopies in Cox Coordinates

Koszul-type determinantal formulas for families of mixed multilinear systems

Yet another eigenvalue algorithm for solving polynomial systems

Contact Info

Product

Resources

About