2020
DOI: 10.48550/arxiv.2006.10654
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Toric Eigenvalue Methods for Solving Sparse Polynomial Systems

Abstract: We consider the problem of computing homogeneous coordinates of points in a zerodimensional subscheme of a compact toric variety X. Our starting point is a homogeneous ideal I in the Cox ring of X, which gives a global description of this subscheme. It was recently shown that eigenvalue methods for solving this problem lead to robust numerical algorithms for solving (nearly) degenerate sparse polynomial systems. In this work, we give a first description of this strategy for non-reduced, zero-dimensional subsch… Show more

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Cited by 4 publications
(17 citation statements)
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“…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%
“…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%
“…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%
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“…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%