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Structure of the inverse to the Sylvester resultant matrix. (English) Zbl 0631.15015
The problem of solving the equation \(ax+by=c\) where a, b and c are given and x and y unknown polynomials of \(\lambda\) of degrees m, n, \(m+n-1\), n- 1, m-1 correspondingly is intimately related to the \((m+n)\times (m+n)\) Sylvester resultant matrix S of polynomials a and b. The author finds a representation of \(S^{-1}\) (if S is regular) as a sum of \(m+n\) dyadic matrices, each of which is a rational function of zeros of a and b. The representation is especially simple if all roots of each of the polynomials a and b are distinct, in which case the representation relates to the Lagrange interpolation formula. In the general case it relates to the general Hermite polynomial interpolation formula.
Reviewer: B.Reichstein
MSC:
15A54 Matrices over function rings in one or more variables
15A09 Theory of matrix inversion and generalized inverses
41A05 Interpolation in approximation theory
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