Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Subalgebras in K[x] of small codimension

Grönkvist, Rode ; Leffler, Erik LU ; Torstensson, Anna LU and Ufnarovski, Victor LU (2022) In Applicable Algebra in Engineering, Communications and Computing 33(6). p.751-789
Abstract

We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in K[x]. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f(α) = f(β) for α, β in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements... (More)

We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in K[x]. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f(α) = f(β) for α, β in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.

(Less)
Please use this url to cite or link to this publication:
author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Derivation, Resultant, SAGBI basis, Subalgebra spectrum
in
Applicable Algebra in Engineering, Communications and Computing
volume
33
issue
6
pages
751 - 789
publisher
Springer
external identifiers
  • scopus:85136576284
ISSN
0938-1279
DOI
10.1007/s00200-022-00573-4
language
English
LU publication?
yes
id
fe5004c9-5551-4d4d-b9f3-86f74605fab9
date added to LUP
2022-10-18 09:16:26
date last changed
2025-04-04 14:46:17
@article{fe5004c9-5551-4d4d-b9f3-86f74605fab9,
  abstract     = {{<p>We introduce the concept of subalgebra spectrum, Sp(A), for a subalgebra A of finite codimension in K[x]. The spectrum is a finite subset of the underlying field. We also introduce a tool, the characteristic polynomial of A, which has the spectrum as its set of zeroes. The characteristic polynomial can be computed from the generators of A, thus allowing us to find the spectrum of an algebra given by generators. We proceed by using the spectrum to get descriptions of subalgebras of finite codimension. More precisely we show that A can be described by a set of conditions that each is either of the type f(α) = f(β) for α, β in Sp(A) or of the type stating that some linear combination of derivatives of different orders evaluated in elements of Sp(A) equals zero. We use these types of conditions to, by an inductive process, find explicit descriptions of subalgebras of codimension up to three. These descriptions also include SAGBI bases for each family of subalgebras.</p>}},
  author       = {{Grönkvist, Rode and Leffler, Erik and Torstensson, Anna and Ufnarovski, Victor}},
  issn         = {{0938-1279}},
  keywords     = {{Derivation; Resultant; SAGBI basis; Subalgebra spectrum}},
  language     = {{eng}},
  number       = {{6}},
  pages        = {{751--789}},
  publisher    = {{Springer}},
  series       = {{Applicable Algebra in Engineering, Communications and Computing}},
  title        = {{Subalgebras in K[x] of small codimension}},
  url          = {{http://dx.doi.org/10.1007/s00200-022-00573-4}},
  doi          = {{10.1007/s00200-022-00573-4}},
  volume       = {{33}},
  year         = {{2022}},
}