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 the j -th unknown in the i -th equation is less than the number of 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 abelianization into an algebraic group G is at least dim G .

中文翻译：

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代数群中方程组的解集维数

Gordon-Rodriguez-Villegas 定理说，在有限群中，如果由 j 的指数和组成的矩阵的秩，则无系数方程组的解的数量可以被群的阶整除 -第 i 个方程中的未知数小于未知数的数量。我们获得了代数群的类似事实和类似事实。特别是，我们的结果意味着，从具有无限阿贝尔化的有限生成群到代数群 G 的各种同态的每个不可约分量的维数至少是暗 G 。