Version 1 of this preprint did not introduce new mathematical ideas, it was more a state of the art June 2021 picture about solving polynomial systems efficiently by reconstructing a rational univariate representation (rur) with a very high probability of correctness using Groebner revlex computation, Berlekamp-Massey algorithm and Hankel linear system solving modulo several primes in parallel.Version 2 introduces an algorithm for rur certification that is effective for most systems (the cost of this algorithmbeing bounded in theorem \ref{prop:check}), something new as far as I know.These algorithms are implemented in \href{https://www-fourier.univ-grenoble-alpes.fr/~parisse/giac.html}{Giac/Xcas} (\cite{giac})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.

翻译：本预印版本1没有引入新的数学理念,更像是2021年6月关于通过重建合理的单词表达法(rur)来有效解决多元体系的艺术状况,该表达法非常可能正确,使用Groebner revlex计算法、Berlekamp-Massey算法和Hankel线性系统同时解决若干质数。Version 2引入了一种对大多数系统有效的rur认证算法(这种算法的费用受Theorem\ref{prop: check}的约束),据我所知,这是新的。 这些算法在\href{https://www-fourier.univ-grenoble-alpes.fr/~parisse/giac.html_Giac/Xcas}(\cite{giac}自1.7.0-13或1.7.7-17版认证以来,它(7月2021日)在多个CPU上,至少是开放源软件上,在多个运行。