Abstract
The space of polynomials in one variable x of odd degree 2n+1 is 2n+2-dimensional. A linear combination of n+1 polynomials (x−β)2n+1 takes care of the required number of parameters. Such an expression has been called the canonical form of a binary form. Sylvester has shown that this reduction is obtained through another polynomial, the catalecticant, when it has the taste of having all its roots distinct. It happens, and this is not by coincidence, that the catalecticant is an orthogonal polynomial for a certain linear functional associated to the polynomial; the required relations are given by the theory of symmetric functions, supplemented by a little amount of divided differences and Lagrange interpolation formula. Having recovered with these tools the classical case, one can go further and treat as well the case where all the roots of the catalecticant are the same; this is what we do in Theorem 3.4.
Keywords
- Orthogonal Polynomial
- Divided Difference
- Lagrange Interpolation Formula
- Sont Nuls
- Relation Suivante
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© 1987 Springer-Verlag
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Lascoux, A. (1987). Forme canonique d'une forme binaire. In: Koh, S.S. (eds) Invariant Theory. Lecture Notes in Mathematics, vol 1278. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0078805
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DOI: https://doi.org/10.1007/BFb0078805
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