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Canonical Bases for Subalgebras on two Generators in the Univariate Polynomial Ring

På svenska

Author

Summary, in English

Abstract. In this paper we examine subalgebras on two generators in the univariate polynomial ring. A set, S, of polynomials in a subalgebra of a polynomial ring is called a canonical basis (also referred to as SAGBI basis) for the subalgebra if all lead monomials in the subalgebra are products of lead monomials of polynomials in S. In this paper we prove that a pair of polynomials ff; gg is a canonical basis for the

subalgebra they generate if and only if both f and g can be written as compositions of polynomials with the same inner polynomial h for some h of degree equal to the greatest common divisor of the degrees of f and g. Especially polynomials of relatively prime degrees constitute a canonical basis. Another special case occurs when the degree of g is a multiple of the degree of f. In this case ff; gg is a canonical basis if

and only if g is a polynomial in f.

Publishing year

2002

Language

English

Pages

565-577

Publication/Series

Beiträge zur Algebra und Geometrie

Volume

43

Issue

2

Document type

Journal article

Publisher

Springer

Topic

  • Mathematics

Keywords

  • canonical bases
  • subalgebra
  • univariate polynomial ring

Status

Published

Research group

  • Algebra

ISBN/ISSN/Other

  • ISSN: 0138-4821