On direct product of algebraic sets over groups

Abstract

We study systems of group equations of the form S = S 1 (X) U S 2 (Y), where X, Y are disjoint sets of variables. The central problem is the description of the radical Rad(S) in terms of the systems S i . We prove that Rad(S) may contain equations which are not derived from equations from Rad(S i ). Systems of equations are considered in the following classes of groups: abelian, free and 2-step nilpotent groups.