448 research outputs found
Deterministically Computing Reduction Numbers of Polynomial Ideals
We discuss the problem of determining reduction number of a polynomial ideal
I in n variables. We present two algorithms based on parametric computations.
The first one determines the absolute reduction number of I and requires
computation in a polynomial ring with (n-dim(I))dim(I) parameters and n-dim(I)
variables. The second one computes via a Grobner system the set of all
reduction numbers of the ideal I and thus in particular also its big reduction
number. However,it requires computations in a ring with n.dim(I) parameters and
n variables.Comment: This new version replaces the earlier version arXiv:1404.1721 and it
has been accepted for publication in the proceedings of CASC 2014, Warsaw,
Polna
Irredundant Triangular Decomposition
Triangular decomposition is a classic, widely used and well-developed way to
represent algebraic varieties with many applications. In particular, there
exist sharp degree bounds for a single triangular set in terms of intrinsic
data of the variety it represents, and powerful randomized algorithms for
computing triangular decompositions using Hensel lifting in the
zero-dimensional case and for irreducible varieties. However, in the general
case, most of the algorithms computing triangular decompositions produce
embedded components, which makes it impossible to directly apply the intrinsic
degree bounds. This, in turn, is an obstacle for efficiently applying Hensel
lifting due to the higher degrees of the output polynomials and the lower
probability of success. In this paper, we give an algorithm to compute an
irredundant triangular decomposition of an arbitrary algebraic set defined
by a set of polynomials in C[x_1, x_2, ..., x_n]. Using this irredundant
triangular decomposition, we were able to give intrinsic degree bounds for the
polynomials appearing in the triangular sets and apply Hensel lifting
techniques. Our decomposition algorithm is randomized, and we analyze the
probability of success
Resource usage analysis of logic programs via abstract interpretation using sized types
We present a novel general resource analysis for logic programs based on sized types. Sized
types are representations that incorporate structural (shape) information and allow expressing
both lower and upper bounds on the size of a set of terms and their subterms at any position
and depth. They also allow relating the sizes of terms and subterms occurring at different
argument positions in logic predicates. Using these sized types, the resource analysis can infer
both lower and upper bounds on the resources used by all the procedures in a program
as functions on input term (and subterm) sizes, overcoming limitations of existing resource
analyses and enhancing their precision. Our new resource analysis has been developed within
the abstract interpretation framework, as an extension of the sized types abstract domain,
and has been integrated into the Ciao preprocessor, CiaoPP. The abstract domain operations
are integrated with the setting up and solving of recurrence equations for inferring both size
and resource usage functions. We show that the analysis is an improvement over the previous
resource analysis present in CiaoPP and compares well in power to state of the art systems
Computing representations for radicals of finitely generated differential ideals
International audienceThis paper deals with systems of polynomial di erential equations, ordinary or with partial derivatives. The embedding theory is the di erential algebra of Ritt and Kolchin. We describe an algorithm, named Rosenfeld-Gröbner, which computes a representation for the radical p of the diff erential ideal generated by any such sys- tem . The computed representation constitutes a normal simpli er for the equivalence relation modulo p (it permits to test embership in p). It permits also to compute Taylor expansions of solutions of . The algorithm is implemented within a package in MAPLE
Solving parametric systems of polynomial equations over the reals through Hermite matrices
We design a new algorithm for solving parametric systems having finitely many
complex solutions for generic values of the parameters. More precisely, let with and
, be the algebraic set
defined by and be the projection . Under the
assumptions that admits finitely many complex roots for generic values of
and that the ideal generated by is radical, we solve the following
problem. On input , we compute semi-algebraic formulas defining
semi-algebraic subsets of the -space such that
is dense in and the number of real points in
is invariant when varies over each .
This algorithm exploits properties of some well chosen monomial bases in the
algebra where is the ideal generated by in
and the specialization property of the so-called Hermite
matrices. This allows us to obtain compact representations of the sets by
means of semi-algebraic formulas encoding the signature of a symmetric matrix.
When satisfies extra genericity assumptions, we derive complexity bounds on
the number of arithmetic operations in and the degree of the
output polynomials. Let be the maximal degree of the 's and , we prove that, on a generic , one can compute
those semi-algebraic formulas with operations in and that the polynomials involved
have degree bounded by .
We report on practical experiments which illustrate the efficiency of our
algorithm on generic systems and systems from applications. It allows us to
solve problems which are out of reach of the state-of-the-art
A general procedure for deriving symmetric expressions for the secant and tangent stiffness matrices in finite element analysis
The paper presents a general and straightforward procedure based on the use of the strain energy density for deriving symmetric expressions of the secant and tangent stiffness matrices for finite element analysis of geometrically non‐linear structural problems. The analogy with previously proposed methods for deriving secant and tangent matrices is detailed. The simplicity of the approach is shown in an example of applicatio
A correct, precise and efficient integration of set-sharing, freeness and linearity for the analysis of finite and rational tree languages
It is well known that freeness and linearity information positively interact with aliasing information, allowing both the precision and the efficiency of the sharing analysis of logic programs to be improved. In this paper, we present a novel combination of set-sharing with freeness and linearity information, which is characterized by an improved abstract unification operator. We provide a new abstraction function and prove the correctness of the analysis for both the finite tree and the rational tree cases.
Moreover, we show that the same notion of redundant information as identified in Bagnara et al. (2000) and Zaffanella et al. (2002) also applies to this abstract domain combination: this allows for the implementation of an abstract unification operator running in polynomial time and achieving the same precision on all the considered observable properties
Solving polynomial constraints for proving termination of rewriting
A termination problem can be transformed into a set of polynomial constraints. Up to now, several approaches have been studied to deal with these constraints as constraint solving problems. In this thesis, we study in depth some of these approaches, present some advances in each approach.Navarro Marset, RA. (2008). Solving polynomial constraints for proving termination of rewriting. http://hdl.handle.net/10251/13626Archivo delegad
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