A proof of the monotone column permanent (MCP) conjecture for dimension 4 via sums-of-squares of rational functions

Abstract

For a proof of the monotone column permanent (MCP)conjecture for dimension 4 it is sufficient to show that 4 polynomials, which come from the permanents of real matrices, are nonnegative for all real values of the variables, where the degrees and the number of the variables of these polynomials are all 8. Here we apply a hybrid symbolic-numerical algorithm for certifying that these polynomials can be written as an exact fraction of two polynomial sums-of-squares (SOS) with rational coefficients.