55 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]
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
A Basic Result on the Theory of Subresultants
Given the polynomials f, g â Z[x] the main result of our paper,
Theorem 1, establishes a direct one-to-one correspondence between the
modified Euclidean and Euclidean polynomial remainder sequences (prsâs) of f, g
computed in Q[x], on one hand, and the subresultant prs of f, g computed
by determinant evaluations in Z[x], on the other.
An important consequence of our theorem is that the signs of Euclidean
and modified Euclidean prsâs - computed either in Q[x] or in Z[x] -
are uniquely determined by the corresponding signs of the subresultant prsâs.
In this respect, all prsâs are uniquely "signed".
Our result fills a gap in the theory of subresultant prsâs. In order to place
Theorem 1 into its correct historical perspective we present a brief historical
review of the subject and hint at certain aspects that need - according to
our opinion - to be revised.
ACM Computing Classification System (1998): F.2.1, G.1.5, I.1.2
On the asymptotic and practical complexity of solving bivariate systems over the reals
This paper is concerned with exact real solving of well-constrained,
bivariate polynomial systems. The main problem is to isolate all common real
roots in rational rectangles, and to determine their intersection
multiplicities. We present three algorithms and analyze their asymptotic bit
complexity, obtaining a bound of \sOB(N^{14}) for the purely projection-based
method, and \sOB(N^{12}) for two subresultant-based methods: this notation
ignores polylogarithmic factors, where bounds the degree and the bitsize of
the polynomials. The previous record bound was \sOB(N^{14}).
Our main tool is signed subresultant sequences. We exploit recent advances on
the complexity of univariate root isolation, and extend them to sign evaluation
of bivariate polynomials over two algebraic numbers, and real root counting for
polynomials over an extension field. Our algorithms apply to the problem of
simultaneous inequalities; they also compute the topology of real plane
algebraic curves in \sOB(N^{12}), whereas the previous bound was
\sOB(N^{14}).
All algorithms have been implemented in MAPLE, in conjunction with numeric
filtering. We compare them against FGB/RS, system solvers from SYNAPS, and
MAPLE libraries INSULATE and TOP, which compute curve topology. Our software is
among the most robust, and its runtimes are comparable, or within a small
constant factor, with respect to the C/C++ libraries.
Key words: real solving, polynomial systems, complexity, MAPLE softwareComment: 17 pages, 4 algorithms, 1 table, and 1 figure with 2 sub-figure
Recursive Polynomial Remainder Sequence and the Nested Subresultants
We give two new expressions of subresultants, nested subresultant and reduced
nested subresultant, for the recursive polynomial remainder sequence (PRS)
which has been introduced by the author. The reduced nested subresultant
reduces the size of the subresultant matrix drastically compared with the
recursive subresultant proposed by the authors before, hence it is much more
useful for investigation of the recursive PRS. Finally, we discuss usage of the
reduced nested subresultant in approximate algebraic computation, which
motivates the present work.Comment: 12 pages. Presented at CASC 2005 (Kalamata, Greece, Septermber 12-16,
2005
On a Theorem by Van Vleck Regarding Sturm Sequences
In 1900 E. B. Van Vleck proposed a very efficient method to compute the Sturm sequence of a polynomial p (x) â Z[x] by triangularizing one of Sylvesterâs matrices of p (x) and its derivative pâČ(x). That method works fine only for the case of complete sequences provided no pivots take place. In 1917, A. J. Pell and R. L. Gordon pointed out this âweaknessâ in
Van Vleckâs theorem, rectified it but did not extend his method, so that it also works in the cases of: (a) complete Sturm sequences with pivot, and (b) incomplete Sturm sequences.
Despite its importance, the Pell-Gordon Theorem for polynomials in Q[x] has been totally forgotten and, to our knowledge, it is referenced by us for the first time in the literature. In this paper we go over Van Vleckâs theorem and method, modify slightly
the formula of the Pell-Gordon Theorem and present a general triangularization method, called the VanVleck-Pell-Gordon method, that correctly computes in Z[x] polynomial Sturm sequences, both complete and incomplete
Subresultants and the Shape Lemma
In nice cases, a zero-dimensional complete intersection ideal over a field of
characteristic zero has a Shape Lemma. There are also cases where the ideal is
generated by the resultant and first subresultant polynomials of the
generators. This paper explores the relation between these representations and
studies when the resultant generates the elimination ideal. We also prove a
Poisson formula for resultants arising from the hidden variable method.Comment: 25 pages, revised version with several changes in sections 2, 3, and
5. Accepted for publication at Mathematics of Computatio
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