A Rational Function Decomposition Algorithm by Near-separated Polynomials

Alonso, César L.; Gutiérrez, Jaime; Recio, Tomás

doi:10.1006/jsco.1995.1030

Cited by 51 publications

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“…The main idea of the present method generalizes one of the univariate rational function decomposition methods presented in Alonso et al (1995) and is based on the nearseparated polynomial concept. This notion was defined only for bivariate polynomials, see also Alonso et al (1997).…”

Section: An Algorithmmentioning

confidence: 98%

“…It is well known that the degree is multiplicative with respect to the composition of univariate rational functions, see Alonso et al (1995). In particular a univariate rational function f ∈ K(x) is a composition unit if there exists g ∈ K(x) such that f (g) = g(f ) = x.…”

Section: Uni-multivariate Rational Decompositionmentioning

confidence: 99%

“…Zippel (1991) presented a polynomial time algorithm to decompose a univariate rational function over any field with efficient polynomial factorization. Alonso et al (1995) presented two exponential-time algorithms to decompose univariate rational functions, which are quite efficient in practice. Klüners (2000) presented an exponential-time algorithm to decompose univariate rational functions over Q.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On Multivariate Rational Function Decomposition

Gutiérrez

Rubio

Sevilla

2002

Journal of Symbolic Computation

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Section: An Algorithmmentioning

confidence: 98%

Section: Uni-multivariate Rational Decompositionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On Multivariate Rational Function Decomposition

Gutiérrez

Rubio

Sevilla

2002

Journal of Symbolic Computation

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“…It is well known that expressing a polynomial as a composition of simpler polynomials has proved useful in attacking diverse computational problems such as evaluating them or finding their roots, providing faster algorithms and allowing proofs that otherwise become more tedious or even infeasible (Alonso et al, 1995;von zur Gathen et al, 2003;Kozen and Landau, 1989). Similar benefits can be expected from expressing polynomial maps as compositions of simpler polynomial maps.…”

Section: Characterize Geometrically the Images Of Polynomial Maps Betmentioning

confidence: 99%

A short proof for the open quadrant problem

Fernando

Ueno

2017

Journal of Symbolic Computation

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In 2003 it was proved that the open quadrant Q := {x > 0, y > 0} of R 2 is a polynomial image of R 2 . This result was the origin of an ulterior more systematic study of polynomial images of Euclidean spaces. In this article we provide a short proof of the previous fact that does not involve computer calculations, in contrast with the original one. The strategy here is to represent the open quadrant as the image of a polynomial map that can be expressed as the composition of three simple polynomial maps whose images can be easily understood.

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“…Note that if r = n, then Q = M. The proof of (2) is analogous to (1). Now let us consider the bases B = {1, .…”

Section: Theorem 31 Let U Be the α-Hypercircle Associated To The Unmentioning

confidence: 99%

Generalizing circles over algebraic extensions

et al. 2009

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Abstract. This paper deals with a family of spatial rational curves that were introduced in 1999 by Andradas, Recio, and Sendra, under the name of hypercircles, as an algorithmic cornerstone tool in the context of improving the rational parametrization (simplifying the coefficients of the rational functions, when possible) of algebraic varieties. A real circle can be defined as the image of the real axis under a Moebius transformation in the complex field. Likewise, and roughly speaking, a hypercircle can be defined as the image of a line ("the K-axis") in an n-degree finite algebraic extension K(α) ≈ K n under the transformationThe aim of this article is to extend, to the case of hypercircles, some of the specific properties of circles. We show that hypercircles are precisely, via K-projective transformations, the rational normal curve of a suitable degree. We also obtain a complete description of the points at infinity of these curves (generalizing the cyclic structure at infinity of circles). We characterize hypercircles as those curves of degree equal to the dimension of the ambient affine space and with infinitely many K-rational points, passing through these points at infinity. Moreover, we give explicit formulae for the parametrization and implicitation of hypercircles. Besides the intrinsic interest of this very special family of curves, the understanding of its properties has a direct application to the simplification of parametrizations problem, as shown in the last section.

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A Rational Function Decomposition Algorithm by Near-separated Polynomials

Cited by 51 publications

References 0 publications

On Multivariate Rational Function Decomposition

On Multivariate Rational Function Decomposition

A short proof for the open quadrant problem

Generalizing circles over algebraic extensions

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