We present a new open source C library msolve dedicated to solving multivariate polynomial systems of dimension zero through computer algebra methods. The core algorithmic framework of msolve relies on Gr{\"o}bner bases and linear algebra based algorithms for polynomial system solving. It relies on Gr{\"o}bner basis computation w.r.t. the degree reverse lexicographical order, Gr{\"o}bner conversion to a lexicographical Gr{\"o}bner basis and real solving of univariate polynomials. We explain in detail how these three main steps of the solving process are implemented, how we exploit AVX2 instruction processors and the more general implementation ideas we put into practice to better exploit the computational capabilities of this algorithmic framework. We compare the practical performances of msolve with leading computer algebra systems such as \textsc{Magma}, \textsc{Maple} , \textsc{Singular} on a wide range of systems with finitely many complex solutions, showing that msolve can tackle systems which were out of reach by the computer algebra software state-of-the-art.

中文翻译：

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msolve：用于求解多项式系统的库

我们提出了一个新的开源C库msolve，该库致力于通过计算机代数方法求解零维的多元多项式系统。msolve的核心算法框架依赖于Gr {\“ o} bner基和基于线性代数的算法来求解多项式系统，它依赖于Gr {\” o} bner基计算，并具有逆字典顺序，即Gr {\“ o } bner转换为词典的Gr {\“ o} bner基础，并真正求解单变量多项式。我们将详细说明解决过程的这三个主要步骤是如何实现的，如何利用AVX2指令处理器，以及为了更好地利用该算法框架的计算能力而付诸实践的更一般的实现思想。