A Note on the Determinant and Permanent Problem

Abstract In Valiant's theory of arithmetic complexity, the following question occupies a central position: Given an integer n, what is the minimal m such that the permanent of an n × n matrix is the projection of the determinant of an m × m matrix? More generally, for affine linear transformations, we ask the similar question: what is the minimal m such that the permanent of an n × n matrix is the determinant of an m × m matrix via affine linear transformation? This paper gives the lower bound m ⩾ ⌊ 2 · n ⌋ , for affine linear transformations. The result is an improvement of earlier results by von zur Gathen, Babai, and Seress. It also generalizes a classical theorem of Marcus and Minc.