DETERMINANT EXPANSIONS OF SIGNED MATRICES AND OF CERTAIN JACOBIANS

This paper treats two topics: Matrices with sign patterns and Jacobians of certain mappings on the nonnegative orthant R(>= 0)(d). The main topic is counting the number of positive and negative coefficients in the determinant expansion of sign patterns and of these Jacobians. The paper is motivated by an approach to chemical networks initiated by Craciun and Feinberg [SIAM J. Appl. Math., 65 (2005), pp. 1526-1546; SIAM J. Appl. Math., 66 (2006), pp. 1321-1338]. We also give a graph-theoretic test for determining when the sign pattern of the Jacobian of a chemical reaction dynamics is unambiguous.