Resultants and Contour Integrals

Resultants are important special functions used to describe nonlinear phenomena. The resultant $R_{r_1\dots r_n}$ determines a consistency condition for a system of $n$ homogeneous polynomials of degrees $r_1,\dots, r_n$ in $n$ variables in precisely the same way as the determinant does for a system of linear equations. Unfortunately, there is a lack of convenient formulas for resultants in the case of a large number of variables. In this paper we use Cauchy contour integrals to obtain a polynomial formula for resultants, which is expected to be useful in applications.