47 research outputs found
Subresultants in multiple roots: an extremal case
We provide explicit formulae for the coefficients of the order-d polynomial
subresultant of (x-\alpha)^m and (x-\beta)^n with respect to the set of
Bernstein polynomials \{(x-\alpha)^j(x-\beta)^{d-j}, \, 0\le j\le d\}. They are
given by hypergeometric expressions arising from determinants of binomial
Hankel matrices.Comment: 18 pages, uses elsart. Revised version accepted for publication at
Linear Algebra and its Application
Subresultants in Multiple Roots
We extend our previous work on Poisson-like formulas for subresultants in
roots to the case of polynomials with multiple roots in both the univariate and
multivariate case, and also explore some closed formulas in roots for
univariate polynomials in this multiple roots setting.Comment: 21 pages, latex file. Revised version accepted for publication in
Linear Algebra and its Application
Symmetric interpolation, Exchange lemma and Sylvester sums.
The theory of symmetric multivariate Lagrange interpolation is a beautiful but rather unknown tool that has many applications. Here we derive from it an Exchange Lemma that allows to explain in a simple and natural way the full description of the double sum expressions introduced by Sylvester in 1853 in terms of subresultants and their Bézout coefficients.Fil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas ; ArgentinaFil: Szanto, Agnes. North Carolina State University; Estados UnidosFil: Valdettaro, Marcelo Alejandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
Sylvester's Double Sums: the general case
In 1853 Sylvester introduced a family of double sum expressions for two
finite sets of indeterminates and showed that some members of the family are
essentially the polynomial subresultants of the monic polynomials associated
with these sets. A question naturally arises: What are the other members of the
family? This paper provides a complete answer to this question. The technique
that we developed to answer the question turns out to be general enough to
charactise all members of the family, providing a uniform method.Comment: 16 pages, uses academic.cls and yjsco.sty. Revised version accepted
for publication in the special issue of the Journal of Symbolic Computation
on the occasion of the MEGA 2007 Conferenc
Subresultants, sylvester sums and the rational interpolation problem
We present a solution for the classical univariate rational interpolation problem by means of (univariate) subresultants. In the case of Cauchy interpolation (interpolation without multiplicities), we give explicit formulas for the solution in terms of symmetric functions of the input data, generalizing the well-known formulas for Lagrange interpolation. In the case of the osculatory rational interpolation (interpolation with multiplicities), we give determinantal expressions in terms of the input data, making explicit some matrix formulations that can independently be derived from previous results by Beckermann and Labahn.Fil: D'Andrea, Carlos. Universidad de Barcelona; EspañaFil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Szanto, Agnes. North Carolina State University; Estados Unido
Closed formula for univariate subresultants in multiple roots
We generalize Sylvester single sums to multisets and show that these sums compute subresultants of two univariate polynomials as a function of their roots independently of their multiplicity structure. This is the first closed formula for subresultants in terms of roots that works for arbitrary polynomials, previous efforts only handled special cases. Our extension involves in some cases confluent Schur polynomials and is obtained by using multivariate symmetric interpolation via an Exchange Lemma.Fil: D'Andrea, Carlos. Universidad de Barcelona; EspañaFil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Szanto, Agnes. North Carolina State University; Estados UnidosFil: Valdettaro, Marcelo Alejandro. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin
An elementary proof of Sylvester's double sums for subresultants
In 1853 Sylvester stated and proved an elegant formula that expresses the polynomial subresultants in terms of the roots of the input polynomials. Sylvester's formula was also recently proved by Lascoux and Pragacz using multi-Schur functions and divided differences. In this paper, we provide an elementary proof that uses only basic properties of matrix multiplication and Vandermonde determinants.Fil: D'Andrea, Carlos. Universidad de Barcelona; EspañaFil: Hong, Hoon. North Carolina State University; Estados UnidosFil: Krick, Teresa Elena Genoveva. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Szanto, Agnes. North Carolina State University; Estados Unido
Sylvester's double sums: an inductive proof of the general case
In 1853 J. Sylvester introduced a family of double sum expressions for two
finite sets of indeterminates and showed that some members of the family are
essentially the polynomial subresultants of the monic polynomials associated
with these sets. In 2009, in a joint work with C. D'Andrea and H. Hong we gave
the complete description of all the members of the family as expressions in the
coefficients of these polynomials. In 2010, M.-F. Roy and A. Szpirglas
presented a new and natural inductive proof for the cases considered by
Sylvester. Here we show how induction also allows to obtain the full
description of Sylvester's double-sums.Comment: 12 pages, uses elsart.cls and yjsco.st