Certifying a probabilistic parallel modular algorithm for rational univariate representation

TL;DR

This paper presents a new certification method for rational univariate representations of polynomial systems, leveraging parallel algorithms and modular computations to enhance correctness verification.

Contribution

It introduces a novel certification approach for rur that is effective for most polynomial systems, implemented in open-source software with high-performance parallel capabilities.

Findings

Effective certification method for rur introduced

Implementation achieves leading performance on multiple CPUs

Applicable to most polynomial systems

Abstract

This paper is about solving polynomial systems. It first recalls how to do that efficiently with a very high probability of correctness by reconstructing a rational univariate representation (rur) using Groebner revlex computation, Berlekamp-Massey algorithm and Hankel linear system solving modulo several primes in parallel. Then it introduces a new method (theorem \ref{prop:check}) for rur certification that is effective for most polynomial systems.These algorithms are implemented in https://www-fourier.univ-grenoble-alpes.fr/~parisse/giac.html since version 1.7.0-13 or 1.7.0-17 for certification, it has (July 2021) leading performances on multiple CPU, at least for an open-source software.