A note on separated factors of separated polynomials
type: Item
instance of: scholarly article
title: A note on separated factors of separated polynomials
zbMATH Open document ID: 0904.12006
DOI: 10.1016/S0022-4049(96)00131-4
author: Cesar Alonso
author: Jaime Gutierrez
author: Tomas Recio
published in: Journal of Pure and Applied Algebra
publication date: 1999-01-19T00:00:00Z
review text: This paper is directed toward decomposition of univariate rational functions [\textit{C. Alonso, J. Gutierrez} and \textit{T. Recio}, FRAC: A Maple package for computing in the rational function field \(\mathbb{K}(X)\), in R. J. Lopez, ed., Maple V: Mathematics and its application (Birkhäuser, Basel), 107-115 (1994; Zbl 0813.65048); A rational function decomposition algorithm by near-separated polynomials, J. Symb. Comput. 19, 527-544 (1995; Zbl 0840.68048); \textit{R. Zippel}, Rational function decomposition, Proc. ISSAC'91 (ACM Press, New York), 1-6 (1991)] by factorizing bivariate polynomials. Let \(\mathbb{K}\) be an arbitrary field. A non-constant polynomial \(f(x,y)\) in \(\mathbb{K} [x,y]\) is called near-separated if there exist non-zero polynomials \(r(x), u(x)\) in \(\mathbb{K}[x]\) and \(s(y), t(y)\) in \(\mathbb{K}[y]\) such that \(r(x)\) (respectively \(s(y)\)) is not a constant times \(u(x)\) (respectively \(t(y)\)) and \(f(x,y)= r(x)s(y)- t(y) u(x)\). The list \([r(x), s(y), t(y), u(x)]\) is called a representation of \(f(x,y)\). The main result extends to rational functions a theorem [\textit{M. Fried} and \textit{R. MacRae}, On curves with separated variables, Math. Ann. 180, 220-226 (1969; Zbl 0185.28803)] concerning separated factors of separated polynomials. Theorem 1. Let \(f(x)= f_n(x)/ f_d(x)\), \(g(x)= g_n(x)/ g_d(x)\), \(f_1(x)= f_{1n}(x)/ f_{1d}(x)\) and \(g_1(x)= g_{1n}(x)/ g_{1d}(x)\) be non-constant rational functions in \(\mathbb{K}(x)\), given in prime form. Then \[ f_{1n}(x) g_{1d}(y)- g_{1n}(y) f_{1d}(x)\quad\text{divides}\quad f_n(x)g_d(y)- g_n(y)f_d(x) \] in \(\mathbb{K}[x,y]\) if and only if there exists a rational function \(h(x)= h_n(x)/ h_d(x)\) in \(\mathbb{K}(x)\), such that \(f(x)= h(f_1(x))\) and \(g(x)= h(g_1(x))\). The proof of the theorem essentially proceeds in one direction by observing that \((x-y)\) divides \(h_n(x)h_d(y)- h_d(x)h_n(y)\) and in the other direction by using the field-theoretic approach of \textit{M. F…
Mathematics Subject Classification ID: 12Y05
Mathematics Subject Classification ID: 26C15
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 13P05
zbMATH DE Number: 1089290
zbMATH Keywords: decomposition of univariate rational functions
zbMATH Keywords: factorizing bivariate polynomials
zbMATH Keywords: separated factors of separated polynomials
reviewed by: Francis Wright
describes a project that uses: FRAC
describes a project that uses: Maple
MaRDI profile type: Publication
cites work: Q4315650
cites work: A rational function decomposition algorithm by near-separated polynomials
cites work: Reconsidering algorithms for real parametric curves
cites work: Polynomial decomposition algorithms
cites work: Q4314358
cites work: On curves with separated variables
cites work: Q3947744
cites work: Q4237364
full work available at URL: s0022-4049(96)00131-4
OpenAlex ID: W2012894434
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description: scientific article; zbMATH DE number 1089290