Solving a fixed number of equations over finite groups

We investigate the complexity of solving systems of polynomial equations over finite groups. In 1999 Goldmann and Russell showed $$\mathrm {NP}$$ 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 $$\mathrm {P}$$ P so far, solving a fixed number of equations is also in $$\mathrm {P}$$ 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.

A new test for asphericity and diagrammatic reducibility of group presentations

We present a new test for studying asphericity and diagrammatic reducibility of group presentations. Our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails. We use this criterion to generalize results of J. Howie and S.M. Gersten on asphericity of LOTs and of Adian presentations, and derive new results on solvability of equations over groups. We also use our methods to investigate a conjecture of S.V. Ivanov related to Kaplansky's problem on zero divisors: we strengthen Ivanov's result for locally indicable groups and prove a weak version of the conjecture.

Non-singular Equations over Groups I

Let G be a group, t an element distinct from G and r(t)= g1tl1 ⋯ gktlk∈ G ∗ 〈t 〉, where each gi is an element of G of order greater than 2 and the li are non-zero integers such that l1+l2+ ⋯ +lk≠ 0 and |li| ≠ |lj| for i ≠ j. It is known that if k≤ 2, then the natural map from G to the one-relator product 〈G,t | r(t)〉 is injective. In this paper, we prove that the same holds for all k ∉ {4, 5}.