Solving a fixed number of equations over finite groups

Abstract

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.