A pr 2 00 1 The Structure of the Inverse to the Sylvester Resultant Matrix

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