Analogue of the identity Log Det =  Trace Log for resultants

Like evaluation of Gaussian integrals is based on determinants, exact (non-perturbative) evaluation of non-Gaussian integrals is related to algebraic quantities called resultants. Resultant Rr1,…,rn defines a condition of solvability for a system of n homogeneous polynomials of degrees r1,…,rn in n variables, just in the same way as a determinant does for a system of linear equations. Because of this, resultants are important special functions of upcoming non-linear physics and begin to play a role in various topics related to string theory. Unfortunately, there is a lack of convenient formulas for resultants when the number of variables is large. To cure this problem, we generalize the well-known identity Log Det = Trace Log from determinants to resultants. The generalized identity allows to obtain explicit polynomial formulas for multidimensional resultants: for any number of variables, the resultant is given by a Schur polynomial. We also give several integral representations for resultants, as well as a sum-over-paths representation.