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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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