Complexity of sparse polynomial solving 3: Infinity

TL;DR

This paper develops a homotopy algorithm for solving sparse polynomial systems over toric varieties, with complexity bounded by a condition number, enabling stable computations near 'toric infinity' and avoiding spurious roots.

Contribution

It introduces a locally defined homotopy algorithm on toric variety charts with complexity bounds based on the condition number, improving stability near infinity.

Findings

Complexity is linearly bounded by the condition length.

The algorithm avoids spurious roots in sparse polynomial solving.

Stable computations are possible near 'toric infinity'.

Abstract

A theory of numerical path-following in toric varieties was suggested in two previous papers. The motivation is solving systems of polynomials with real or complex coefficients. When those polynomials are not assumed 'dense', solving them over projective space or complex space may introduce spurious, degenerate roots or components. Spurious roots may be avoided by solving over toric varieties. In this paper, a homotopy algorithm is locally defined on charts of the toric variety. Its complexity is bounded linearly by the condition length, that is the integral along the lifted path (coefficients and solution) of thetoric condition number. Those charts allow for stable computations near "toric infinity",which was not possible within the technology of the previous papers.