The dimension of solution sets to systems of equations in algebraic groups

The Gordon--Rodriguez-Villegas theorem says that, in a finite group, the number of solutions to a system of coefficient-free equations is divisible by the order of the group if the rank of the matrix composed of the exponent sums of $j$-th unknown in $i$-th equation is less than the number unknowns. We obtain analogues of this and similar facts for algebraic groups. In particular, our results imply that the dimension of each irreducible component of the variety of homomorphisms from a finitely generated group with infinite abelianisation into an algebraic group $G$ is at least $\dim G$.