We investigate the complexity of solving systems of polynomial equations over finite groups. In 1999 Goldmann and Russell showed \(\mathrm {NP}\)-completeness of this problem for non-Abelian groups. We show that the problem can become tractable for some non-Abelian groups if we fix the number of equations. Recently, Földvári and Horváth showed that a single equation over groups which are semidirect products of a p-group with an Abelian group can be solved in polynomial time. We generalize this result and show that the same is true for systems with a fixed number of equations. This shows that for all groups for which the complexity of solving one equation has been proved to be in \(\mathrm {P}\) so far, solving a fixed number of equations is also in \(\mathrm {P}\). Using the collecting procedure presented by Horváth and Szabó in 2006, we furthermore present a faster algorithm to solve systems of equations over groups of order pq.

我们研究了有限组上多项式方程组求解系统的复杂性。1999年，戈德曼和罗素证明了\（\ mathrm {NP} \）-对于非阿贝尔族而言，该问题的完备性。我们证明，如果我们固定方程组的数量，那么对于某些非阿贝尔群体而言，该问题可能变得易于解决。最近，Földvári和Horváth表明，可以在多项式时间内求解p组与Abelian组的半直接乘积的单个方程组。我们对这个结果进行了概括，并表明对于方程数目固定的系统也是如此。这表明，到目前为止，对于已经证明求解一个方程的复杂度在\（\ mathrm {P} \）中的所有组，求解固定数量的方程也是\（\ mathrm {P} \）。使用Horváth和Szabó在2006年提出的收集程序，我们进一步提出了一种更快的算法来求解阶数pq上的方程组 。