215 research outputs found
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
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
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
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
Improved Polynomial Remainder Sequences for Ore Polynomials
Polynomial remainder sequences contain the intermediate results of the
Euclidean algorithm when applied to (non-)commutative polynomials. The running
time of the algorithm is dependent on the size of the coefficients of the
remainders. Different ways have been studied to make these as small as
possible. The subresultant sequence of two polynomials is a polynomial
remainder sequence in which the size of the coefficients is optimal in the
generic case, but when taking the input from applications, the coefficients are
often larger than necessary. We generalize two improvements of the subresultant
sequence to Ore polynomials and derive a new bound for the minimal coefficient
size. Our approach also yields a new proof for the results in the commutative
case, providing a new point of view on the origin of the extraneous factors of
the coefficients
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
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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