Canonical Bases for Subalgebras on two Generators in the Univariate Polynomial Ring
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Abstract
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.
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.
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Research areas and keywords  Subject classification (UKÄ) – MANDATORY
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Original language  English 

Pages (fromto)  565577 
Journal  Beiträge zur Algebra und Geometrie 
Volume  43 
Issue number  2 
Publication status  Published  2002 
Publication category  Research 
Peerreviewed  Yes 