Certifying zeros of polynomial systems using interval arithmetic

TL;DR

This paper introduces a practical interval arithmetic-based method for certifying solutions to polynomial systems, integrated into HomotopyContinuation.jl, significantly improving certification efficiency in numerical algebraic geometry.

Contribution

It presents a new certification method using interval arithmetic and Krawczyk's method, integrated into existing software, enhancing reliability and performance.

Findings

The certify function reliably proves correctness of solutions.

The new approach outperforms previous certification methods.

Certification can become standard practice in numerical algebraic geometry.

Abstract

We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated nonsingular solution to a square system of polynomial equations. The implementation rests on Krawczyk's method. We demonstrate that it dramatically outperforms earlier approaches to certification. We see this contribution as powerful new tool in numerical algebraic geometry, that can make certification the default and not just an option.