An upper bound on the number of monomials in determinants of sparse matrices with symbolic entries

The objective of this paper is to gain some insight into how well sparsity is preserved under determinant computations. For a square matrix A whose elements are indeterminates x1, . . . , xn and zeros, the determinant det(A) is a polynomial in x1, . . . , xn with integer coefficients. We derive an upper bound on the number of monomials in det(A) for a class of determinants which includes bigradients, Sylvester resultants and determinants of Toeplitz and Hankel matrices. Our approach is based on a result by Stanley in the theory of partially ordered sets.