An identity for bipartite matching and symmetric determinant

Abstract

Let A( x) = ( A ij ( x)) be an n × n symmetric polynomial matrix in x, and define δ( A) = deg x det A( x), [the degree of the determinant of A( x)]. Let G( A) be the associated bipartite graph; the vertices of G( A) are identified with the rows and the columns of A, and the edges with the nonzero entries of A. We attach deg x A ij ( x) to edge ( i,j) of G( A) as a weight and define \\ ̂ gd 0(A) to be the maximum weight of a perfect matching in G( A). Then δ(A) ≤ \\ ̂ gd 0(A) , and the equality holds in the generic case, i.e., when the leading coefficients of A ij ( x) ( i ≤ j) are independent parameters. It is proven by a combinatorial argument that the gap between δ( A) and \\ ̂ gd 0(A) , if any, can be resolved by a unimodular congruence transformation. That is, for a nonsingular A( x) there exists a unimodular U( x) such that A′( x) = U( x) A( x) U( x) T satisfies (δ(A) =) δ(A′) = \\ ̂ gd 0(A′) . Note that if A( x) is transformed to A′( x) = U( x) A( x) U( x) T, δ( A) remains invariant, whereas \\ ̂ gd 0(A) does change, since G( A) changes. The proof relies on the dual integrality of bipartite matching polytopes well known in polyhedral combinatorics.