We study boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. We prove that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori. Here balanced means that the size of the torus is at most exponential in the shortest side length, while highly unbalanced means that the longest side length of the torus is super exponential in the product over the other side lengths cubed. We discuss implications of our results to the existence of efficient algorithms for approximating the independence polynomial on tori.

翻译：我们研究收敛于整数格点的环面序列的独立多项式零点的有界性。我们证明，对于平衡环面序列，零点是有界的；而对于高度非平衡环面序列，零点是无界的。这里"平衡"是指环面的大小至多为最短边长的指数函数，而"高度非平衡"是指环面的最长边长超过其他边长乘积的立方项的指数函数。我们讨论了研究结果对于在环面上近似独立多项式的高效算法存在性的启示。