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Abstract

Algebraic sets play a central role in algebraic geometry over groups; by definition, an algebraic set is the set of common roots of a collection of equations in some non-commuting variables x1, ..., xn having coefficients in some group G. Often, one seeks solutions of equations over G in the affine n-space Gn; such systems of polynomial equations are termed Diophantine systems. In this brief note, we consider Diophantine systems over a group G that is assumed to be nilpotent of class c and to have an element of infinite order in its center; we associate with each n × n invertible matrix A having integral coefficients an algebraic set YAac and prove that (γc(G))n ⊆ YAac; here, γc(G) stands for the subgroup of G generated by all simple commutators of length c + 1. Consequently, we prove that if G is an abelian group having an element of infinite order, then there exist infinitely many subsets of the affine n-space Gn having cardinality n + 1 that are algebraically dense in Gn; in particular, this provides us with another proof of the well-known fact that the union of two algebraic sets need not be an algebraic set.

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