Are There Elimination Algorithms for the Permanent?

Abstract

We define the class of elimination algorithms. These are algebraic algorithms for evaluating multivariate polynomials, and include as a special case, Gaussian elimination for evaluating the determinant. We show how to find the linear symmetries of a polynomial, defined appropriately, and use these methods to find the linear symmetries of the permanent and determinant. We show that in contrast to Gaussian elimination for the determinant, there is no elimination algorithm for the permanent.