  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 $\dim G$.
