123 research outputs found
Symmetric Subresultants and Applications
Schur's transforms of a polynomial are used to count its roots in the unit
disk. These are generalized them by introducing the sequence of symmetric
sub-resultants of two polynomials. Although they do have a determinantal
definition, we show that they satisfy a structure theorem which allows us to
compute them with a type of Euclidean division. As a consequence, a fast
algorithm based on a dichotomic process and FFT is designed. We prove also that
these symmetric sub-resultants have a deep link with Toeplitz matrices.
Finally, we propose a new algorithm of inversion for such matrices. It has the
same cost as those already known, however it is fraction-free and consequently
well adapted to computer algebra
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
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 CientÃficas 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 CientÃficas y Técnicas; Argentin
Multivariate Subresultants in Roots
We give rational expressions for the subresultants of n+1 generic polynomials
f_1,..., f_{n+1} in n variables as a function of the coordinates of the common
roots of f_1,..., f_n and their evaluation in f_{n+1}. We present a simple
technique to prove our results, giving new proofs and generalizing the
classical Poisson product formula for the projective resultant, as well as the
expressions of Hong for univariate subresultants in roots.Comment: 22 pages, no figures, elsart style, revised version of the paper
presented in MEGA 2005, accepted for publication in Journal of Algebr
On principal minors of Bezout matrix
Let be real numbers, , and
be a polynomial of degree less than or equal to . Denote by the
matrix of generalized divided differences of with nodes
and by the Bezout matrix (Bezoutiant) of and . A relationship
between the corresponding principal minors, counted from the right-hand lower
corner, of the matrices and is established. It implies
that if the principal minors of the matrix of divided differences of a function
are positive or have alternating signs then the roots of the Newton's
interpolation polynomial of are real and separated by the nodes of
interpolation.Comment: 15 page
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
Computing greatest common divisor of several parametric univariate polynomials via generalized subresultant polynomials
In this paper, we tackle the following problem: compute the gcd for several
univariate polynomials with parametric coefficients. It amounts to partitioning
the parameter space into ``cells'' so that the gcd has a uniform expression
over each cell and constructing a uniform expression of gcd in each cell. We
tackle the problem as follows. We begin by making a natural and obvious
extension of subresultant polynomials of two polynomials to several
polynomials. Then we develop the following structural theories about them.
1. We generalize Sylvester's theory to several polynomials, in order to
obtain an elegant relationship between generalized subresultant polynomials and
the gcd of several polynomials, yielding an elegant algorithm.
2. We generalize Habicht's theory to several polynomials, in order to obtain
a systematic relationship between generalized subresultant polynomials and
pseudo-remainders, yielding an efficient algorithm.
Using the generalized theories, we present a simple (structurally elegant)
algorithm which is significantly more efficient (both in the output size and
computing time) than algorithms based on previous approaches
New and Old Results in Resultant Theory
Resultants are getting increasingly important in modern theoretical physics:
they appear whenever one deals with non-linear (polynomial) equations, with
non-quadratic forms or with non-Gaussian integrals. Being a subject of more
than three-hundred-year research, resultants are of course rather well studied:
a lot of explicit formulas, beautiful properties and intriguing relationships
are known in this field. We present a brief overview of these results,
including both recent and already classical. Emphasis is made on explicit
formulas for resultants, which could be practically useful in a future physics
research.Comment: 50 pages, 15 figure
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 CientÃficas 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 CientÃficas y Técnicas; Argentin
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