410 research outputs found
Recursive Polynomial Remainder Sequence and its Subresultants
We introduce concepts of "recursive polynomial remainder sequence (PRS)" and
"recursive subresultant," along with investigation of their properties. A
recursive PRS is defined as, if there exists the GCD (greatest common divisor)
of initial polynomials, a sequence of PRSs calculated "recursively" for the GCD
and its derivative until a constant is derived, and recursive subresultants are
defined by determinants representing the coefficients in recursive PRS as
functions of coefficients of initial polynomials. We give three different
constructions of subresultant matrices for recursive subresultants; while the
first one is built-up just with previously defined matrices thus the size of
the matrix increases fast as the recursion deepens, the last one reduces the
size of the matrix drastically by the Gaussian elimination on the second one
which has a "nested" expression, i.e. a Sylvester matrix whose elements are
themselves determinants.Comment: 30 pages. Preliminary versions of this paper have been presented at
CASC 2003 (arXiv:0806.0478 [math.AC]) and CASC 2005 (arXiv:0806.0488
[math.AC]
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
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
Subresultants and Generic Monomial Bases
Given n polynomials in n variables of respective degrees d_1,...,d_n, and a
set of monomials of cardinality d_1...d_n, we give an explicit
subresultant-based polynomial expression in the coefficients of the input
polynomials whose non-vanishing is a necessary and sufficient condition for
this set of monomials to be a basis of the ring of polynomials in n variables
modulo the ideal generated by the system of polynomials. This approach allows
us to clarify the algorithms for the Bezout construction of the resultant.Comment: 22 pages, uses elsart.cls. Revised version accepted for publication
in the Journal of Symbolic Computatio
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
Resultants and subresultants of p-adic polynomials
We address the problem of the stability of the computations of resultants and
subresultants of polynomials defined over complete discrete valuation rings
(e.g. Zp or k[[t]] where k is a field). We prove that Euclide-like algorithms
are highly unstable on average and we explain, in many cases, how one can
stabilize them without sacrifying the complexity. On the way, we completely
determine the distribution of the valuation of the principal subresultants of
two random monic p-adic polynomials having the same degree
On the irreducibility of multivariate subresultants
Let be generic homogeneous polynomials in variables of
degrees respectively. We prove that if is an integer
satisfying then all multivariate
subresultants associated to the family in degree are
irreducible. We show that the lower bound is sharp. As a byproduct, we get a
formula for computing the residual resultant of
smooth isolated points in \PP^{n-1}.Comment: Updated version, 4 pages, to appear in CRA
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 by 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.Comment: 9 pages, no figures, simpler proof of the main results thanks to
useful comments made by the referees. To appear in Journal of Symbolic
Computatio
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