52 research outputs found
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
Over-constrained Weierstrass iteration and the nearest consistent system
We propose a generalization of the Weierstrass iteration for over-constrained
systems of equations and we prove that the proposed method is the Gauss-Newton
iteration to find the nearest system which has at least common roots and
which is obtained via a perturbation of prescribed structure. In the univariate
case we show the connection of our method to the optimization problem
formulated by Karmarkar and Lakshman for the nearest GCD. In the multivariate
case we generalize the expressions of Karmarkar and Lakshman, and give
explicitly several iteration functions to compute the optimum.
The arithmetic complexity of the iterations is detailed
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
On deflation and multiplicity structure
This paper presents two new constructions related to singular solutions of
polynomial systems. The first is a new deflation method for an isolated
singular root. This construction uses a single linear differential form defined
from the Jacobian matrix of the input, and defines the deflated system by
applying this differential form to the original system. The advantages of this
new deflation is that it does not introduce new variables and the increase in
the number of equations is linear in each iteration instead of the quadratic
increase of previous methods. The second construction gives the coefficients of
the so-called inverse system or dual basis, which defines the multiplicity
structure at the singular root. We present a system of equations in the
original variables plus a relatively small number of new variables that
completely deflates the root in one step. We show that the isolated simple
solutions of this new system correspond to roots of the original system with
given multiplicity structure up to a given order. Both constructions are
"exact" in that they permit one to treat all conjugate roots simultaneously and
can be used in certification procedures for singular roots and their
multiplicity structure with respect to an exact rational polynomial system.Comment: arXiv admin note: substantial text overlap with arXiv:1501.0508
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
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
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