The Structure of the Inverse to the Sylvester Resultant Matrix

Abstract

Given polynomials a(z) of degree m and b(z) of degree n, we represent the inverse to the Sylvester resultant matrix of a(z) and b(z), if this inverse exists, as a canonical sum of m+n dyadic matrices each of which is a rational function of zeros of a(z) and b(z). As a result, we obtain the polynomial solutions X(z) of degree n-1 and Y(z) of degree m-1 to the equation a(z)X(z)+b(z)Y(z)=c(z), where c(z) is a given polynomial of degree m+n-1, as follows: X(z) is a Lagrange interpolation polynomial for the function c(z)/a(z) over the set of zeros of b(z) and Y(z) is the one for the function c(z)/b(z) over the set of zeros of a(z).