Verified Error Bounds for Isolated Singular Solutions of Polynomial Systems

TL;DR

This paper introduces a new verification method using deflation and smoothing parameters to compute guaranteed, narrow error bounds for isolated singular solutions of polynomial systems, even with high multiplicity.

Contribution

It generalizes previous algorithms to handle broader cases and provides a practical approach for certifying singular solutions with high multiplicity.

Findings

Effective for systems with multiplicity up to hundreds

Produces verified, narrow error bounds

Applicable to a wide range of polynomial systems

Abstract

In this paper, we generalize the algorithm described by Rump and Graillat, as well as our previous work on certifying breadth-one singular solutions of polynomial systems, to compute verified and narrow error bounds such that a slightly perturbed system is guaranteed to possess an isolated singular solution within the computed bounds. Our new verification method is based on deflation techniques using smoothing parameters. We demonstrate the performance of the algorithm for systems with singular solutions of multiplicity up to hundreds.