Subalgebras of the univariate polynomial algebra
(2026) In Applicable Algebra in Engineering, Communications and Computing- Abstract
In previous work we have shown that any subalgebra A⊆K[x] of finite codimension n can be described by a finite set Sp(A), called the subalgebra spectrum of A, together with n defining conditions. In this paper we show that a similar result applies to subalgebras of infinite codimension. We also develop new tools for the case of finite codimension. One of them is to introduce an ideal I(A) in A of finite codimension in K[x], such that V(I(A))=Sp(A). We use A/I(A) to analyse how the structure of A relates to its defining conditions. A key problem is to find algorithms for obtaining a SAGBI basis given defining conditions and vice versa. We present an efficient solution to the first problem. By elimination of a linear system we obtain a... (More)
In previous work we have shown that any subalgebra A⊆K[x] of finite codimension n can be described by a finite set Sp(A), called the subalgebra spectrum of A, together with n defining conditions. In this paper we show that a similar result applies to subalgebras of infinite codimension. We also develop new tools for the case of finite codimension. One of them is to introduce an ideal I(A) in A of finite codimension in K[x], such that V(I(A))=Sp(A). We use A/I(A) to analyse how the structure of A relates to its defining conditions. A key problem is to find algorithms for obtaining a SAGBI basis given defining conditions and vice versa. We present an efficient solution to the first problem. By elimination of a linear system we obtain a linear basis of A and then reduce it into a minimal SAGBI basis. By solving the same linear system we can also obtain the normal form of any polynomial. Further, we suggest an efficient way to obtain Sp(A) from generators of A, but note that finding the conditions is NP hard.
(Less)
- author
- Torstensson, Anna LU
- organization
- publishing date
- 2026
- type
- Contribution to journal
- publication status
- epub
- subject
- keywords
- Defining conditions, Polynomial subalgebra, SAGBI basis, Subalgebra spectrum
- in
- Applicable Algebra in Engineering, Communications and Computing
- publisher
- Springer
- external identifiers
-
- scopus:105027887970
- ISSN
- 0938-1279
- DOI
- 10.1007/s00200-025-00724-3
- language
- English
- LU publication?
- yes
- id
- 9ba886f5-d837-4c15-9581-996504c9fcb0
- date added to LUP
- 2026-02-25 15:48:10
- date last changed
- 2026-02-25 15:49:04
@article{9ba886f5-d837-4c15-9581-996504c9fcb0,
abstract = {{<p>In previous work we have shown that any subalgebra A⊆K[x] of finite codimension n can be described by a finite set Sp(A), called the subalgebra spectrum of A, together with n defining conditions. In this paper we show that a similar result applies to subalgebras of infinite codimension. We also develop new tools for the case of finite codimension. One of them is to introduce an ideal I(A) in A of finite codimension in K[x], such that V(I(A))=Sp(A). We use A/I(A) to analyse how the structure of A relates to its defining conditions. A key problem is to find algorithms for obtaining a SAGBI basis given defining conditions and vice versa. We present an efficient solution to the first problem. By elimination of a linear system we obtain a linear basis of A and then reduce it into a minimal SAGBI basis. By solving the same linear system we can also obtain the normal form of any polynomial. Further, we suggest an efficient way to obtain Sp(A) from generators of A, but note that finding the conditions is NP hard.</p>}},
author = {{Torstensson, Anna}},
issn = {{0938-1279}},
keywords = {{Defining conditions; Polynomial subalgebra; SAGBI basis; Subalgebra spectrum}},
language = {{eng}},
publisher = {{Springer}},
series = {{Applicable Algebra in Engineering, Communications and Computing}},
title = {{Subalgebras of the univariate polynomial algebra}},
url = {{http://dx.doi.org/10.1007/s00200-025-00724-3}},
doi = {{10.1007/s00200-025-00724-3}},
year = {{2026}},
}