### 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:
https://lup.lub.lu.se/record/954419

- author
- Torstensson, Anna
^{LU} - organization
- publishing date
- 2002
- 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}}, }