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

(2002) In Beiträge zur Algebra und Geometrie 43(2). p.565-577
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... (More)
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. (Less)
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
canonical bases, subalgebra, univariate polynomial ring
in
Beiträge zur Algebra und Geometrie
volume
43
issue
2
pages
565 - 577
publisher
Springer
ISSN
0138-4821
language
English
LU publication?
yes
id
c263497a-3765-41f0-a16c-896bbcadd241 (old id 954419)
alternative location
http://www.math.kth.se/~annator/SAGBI.pdf
date added to LUP
2016-04-01 11:51:42
date last changed
2023-10-13 11:44:58
```@article{c263497a-3765-41f0-a16c-896bbcadd241,
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<br/><br>
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<br/><br>
and only if g is a polynomial in f.}},
author       = {{Torstensson, Anna}},
issn         = {{0138-4821}},
keywords     = {{canonical bases; subalgebra; univariate polynomial ring}},
language     = {{eng}},
number       = {{2}},
pages        = {{565--577}},
publisher    = {{Springer}},
series       = {{Beiträge zur Algebra und Geometrie}},
title        = {{Canonical Bases for Subalgebras on two Generators in the Univariate Polynomial Ring}},
url          = {{http://www.math.kth.se/~annator/SAGBI.pdf}},
volume       = {{43}},
year         = {{2002}},
}

```