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arxiv 2106.10912 v3 pith:QHHBW6K3 submitted 2021-06-21 cs.SC

Certifying a probabilistic parallel modular algorithm for rational univariate representation

classification cs.SC
keywords algorithmcertificationparallelpolynomialrationalrepresentationsolvingsystems
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved
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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.

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  1. Fast Rational Univariate Representation via Gaussian Elimination

    cs.SC 2026-07 conditional novelty 4.0

    A Julia package using dense Gaussian elimination computes certified rational univariate representations of polynomial systems with thousands of solutions in seconds, outperforming msolve on several benchmarks.