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Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry, Vol. 43, No. 2, pp. 565-577 (2002)
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Canonical Bases for Subalgebras on Two Generators in the Univariate Polynomial Ring

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Anna Torstensson

Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden, e-mail: annat@maths.lth.se

**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 $\{f,g\}$ 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 $\{f,g\}$ is a canonical basis if and only if $g$ is a polynomial in $f$.

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