Basic algorithms for rational function fields

Название: Basic algorithms for rational function fields
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Категория: Математика
Тип: Книга
Дата: 30.12.2008 15:59:34
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Описание: By means of Grobner basis techniques algorithms for solving various problems concerning subfields к (g) := к (gi,..., gm) of a rational function field к (x) := к (xi, ■ ■ ■, xn) are derived: computing canonical generating sets, deciding field membership, computing the degree and separability degree resp. the transcendence degree and a transcendence basis of K(x)/K(g), deciding whether / S к(х) is algebraic or transcendental over K(g), computing minimal polynomials, and deciding whether к (g) contains elements of a "particular structure", e.g. monic univariate polynomials of fixed degree. The essential idea is to reduce these problems to questions concerning an ideal of a polynomial ring; connections between minimal primary decompositions over к (х) of this ideal and intermediate fields of к (g) and к (х) are given. In the last section some practical considerations concerning the use of the algorithms are discussed. © 1999 Academic Press 0. Introduction Rational function fields K(x) := K(rri,... ,xn) arise in various contexts within mathematics and computer science. Two examples are invariant theory (Kemper, 1994) and the design of diffractive optical systems (Aagedal et al., 1996). In the latter, multivariate decomposition of rational functions proves useful for inverting rational functions. For performing practical calculations in subfields K(g) := K(<7i,... ,gm) of K(x) it is desirable to have algorithms for solving problems like field membership or the computation of a canonical generating set. The algorithms given in Sweedler (1993) and Kemper (1993) are mainly concerned with questions like calculating minimal polynomials over K(g) and finding the transcendental/algebraic degree of an extension K(x)/K(g). In addition to these algorithms for deciding the field membership problem are given. Both Sweedler and Kemper make use of Grobner basis techniques with the introduction of additional ("tag") variables and a lexicographical order of the terms in x, leading to difficulties in practical computations. The motivation of this paper on the one hand is to find alternate solutions to the problems discussed by Sweedler and Kemper in order to broaden the spectrum of effectively tractable problems; on the other hand, we want to give new algorithms for questions concerning subfields K(g). For this we associate to K(g) an ideal in the polynomial ring K(g)[Z], thereby reducing problems such as computing a canonical generating set of K(g) over IK or determining the type of the extension K(x)/K(g) to problems concerning an ideal in a polynomial ring. More precisely the problems treated in the text are: 0747-7171/99/020143 + 28 $30.00/0 © 1999 Academic Press
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