Bipartite perfect matching as a real polynomial

We obtain a description of the Bipartite Perfect Matching decision problem as a multilinear polynomial over the Reals. We show that it has full total degree and (1−o(1))⋅2n2 monomials with non-zero coefficients. In contrast, we show that in the dual representation (switching the roles of 0 and 1) the number of monomials is only exponential in Θ(n log n). Our proof relies heavily on the fact that the lattice of graphs which are “matching-covered” is Eulerian.