Numerical root finding via Cox rings

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:224

Article

Telen, Simon¹ 

[1] Katholieke Univ Leuven, Dept Comp Sci, Celestijnenlaan 200a, B-3001 Heverlee, Belgium

关键词: Cox ring;    Systems of polynomial equations;    Toric varieties;    Multiplication matrix;   

DOI  :  10.1016/j.jpaa.2020.106367

来源: Elsevier

【 摘 要 】

We present a new eigenvalue method for solving a system of Laurent polynomial equations defining a zero-dimensional reduced subscheme of a toric compactification X of (C\{0})(n). We homogenize the input equations to obtain a homogeneous ideal I in the Cox ring of X and generalize the eigenvalue, eigenvector theorem for root finding in affine space to compute homogeneous coordinates of the solutions. Several numerical experiments show the effectiveness of the resulting method. In particular, the method outperforms existing solvers in the case of (nearly) degenerate systems with solutions on or near the torus invariant prime divisors. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free