Numerical Root Finding via Cox Rings

TL;DR

This paper introduces a novel eigenvalue-based method for solving systems of Laurent polynomial equations using Cox rings, improving efficiency especially for degenerate systems near torus invariant divisors.

Contribution

It generalizes the eigenvalue method to homogeneous ideals in Cox rings, enabling effective rootfinding in toric compactifications.

Findings

Outperforms existing solvers on degenerate systems

Effective for solutions near torus invariant divisors

Demonstrated through multiple numerical experiments

Abstract

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 $(\mathbb{C} \setminus \{0\})^n$. We homogenize the input equations to obtain a homogeneous ideal in the Cox ring of $X$ and generalize the eigenvalue, eigenvector theorem for rootfinding 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.