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Describing multivariate polynomial subalgebras using equations

Leffler, Erik LU (2026) In Applicable Algebra in Engineering, Communications and Computing
Abstract

In this paper we continue the work of describing polynomial subalgebras of finite codimension that was started in Grönkvist et al. (Appl Algebra Eng Commun Comput 33(6):751–789, 2022). Let K be an algebraically closed field, and A⊂K[x1,…,xn] be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of K-algebras (Formula presented.) where each Ai can be written as the kernel of some linear functional Li+1:Ai+1→K, and each Li is either a derivation or of the form Li:f→c(f(α)-f(β)) for some α,β∈Kn and c∈K. We investigate the structure of these filtrations and linear functionals. Our main result shows that each such Li which is a derivation may be written as a linear combination of... (More)

In this paper we continue the work of describing polynomial subalgebras of finite codimension that was started in Grönkvist et al. (Appl Algebra Eng Commun Comput 33(6):751–789, 2022). Let K be an algebraically closed field, and A⊂K[x1,…,xn] be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of K-algebras (Formula presented.) where each Ai can be written as the kernel of some linear functional Li+1:Ai+1→K, and each Li is either a derivation or of the form Li:f→c(f(α)-f(β)) for some α,β∈Kn and c∈K. We investigate the structure of these filtrations and linear functionals. Our main result shows that each such Li which is a derivation may be written as a linear combination of partial derivatives evaluated at points of Kn.

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Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
Defining conditions, Derivation, Polynomial subalgebra, Subalgebra spectrum
in
Applicable Algebra in Engineering, Communications and Computing
publisher
Springer
external identifiers
  • scopus:105042221333
ISSN
0938-1279
DOI
10.1007/s00200-026-00743-8
language
English
LU publication?
yes
id
40aef394-31c2-4ef6-a682-04dac681eb5c
date added to LUP
2026-06-29 14:12:17
date last changed
2026-06-29 14:13:19
@article{40aef394-31c2-4ef6-a682-04dac681eb5c,
  abstract     = {{<p>In this paper we continue the work of describing polynomial subalgebras of finite codimension that was started in Grönkvist et al. (Appl Algebra Eng Commun Comput 33(6):751–789, 2022). Let K be an algebraically closed field, and A⊂K[x1,…,xn] be a subalgebra of finite codimension. It is known that there exists a (not necessarily unique) finite filtration of K-algebras (Formula presented.) where each Ai can be written as the kernel of some linear functional Li+1:Ai+1→K, and each Li is either a derivation or of the form Li:f→c(f(α)-f(β)) for some α,β∈Kn and c∈K. We investigate the structure of these filtrations and linear functionals. Our main result shows that each such Li which is a derivation may be written as a linear combination of partial derivatives evaluated at points of Kn.</p>}},
  author       = {{Leffler, Erik}},
  issn         = {{0938-1279}},
  keywords     = {{Defining conditions; Derivation; Polynomial subalgebra; Subalgebra spectrum}},
  language     = {{eng}},
  publisher    = {{Springer}},
  series       = {{Applicable Algebra in Engineering, Communications and Computing}},
  title        = {{Describing multivariate polynomial subalgebras using equations}},
  url          = {{http://dx.doi.org/10.1007/s00200-026-00743-8}},
  doi          = {{10.1007/s00200-026-00743-8}},
  year         = {{2026}},
}