# LUP Student Papers

## LUND UNIVERSITY LIBRARIES

### Describing Finite Codimensional Polynomial Subalgebras Using Partial Derivatives

(2022) In Master's Theses in Mathematical Sciences FMAM05 20221
Mathematics (Faculty of Engineering)
Abstract
In this text we continue the work of describing subalgebras of K[x] of ﬁnite codimension that was started in Describing subalgebras of K[x] using derivatives. In the referenced paper, the authors present how univariate subalgebras can be described by conditions based on evaluations in certain scalars, and proceed to develop a large theoretical framework to understand the nature of such conditions. The purpose of this thesis is to generalize as many of their results as possible to the multivariate setting K[x]. We include generalized deﬁnitions of the type, spectrum, clusters, α-derivations, and α, β-evaluation subtractions. We also state and prove generalizations of most of the theorems relating to the spectrum, clusters, α-derivation... (More)
In this text we continue the work of describing subalgebras of K[x] of ﬁnite codimension that was started in Describing subalgebras of K[x] using derivatives. In the referenced paper, the authors present how univariate subalgebras can be described by conditions based on evaluations in certain scalars, and proceed to develop a large theoretical framework to understand the nature of such conditions. The purpose of this thesis is to generalize as many of their results as possible to the multivariate setting K[x]. We include generalized deﬁnitions of the type, spectrum, clusters, α-derivations, and α, β-evaluation subtractions. We also state and prove generalizations of most of the theorems relating to the spectrum, clusters, α-derivation spaces, as well as the Main Theorem. We also give a couple of new results pertaining to how α-derivation spaces behave when we apply subalgebra conditions on clusters not containing α. (Less)
Popular Abstract
Mathematicians like to study different kinds of structures, and this thesis is concerned with one structure in particular, Polynomial Algebras. The main purpose of the thesis is to develop a new method for describing such algebras; by using of conditions.
author
supervisor
organization
course
FMAM05 20221
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Algebra Polynomial Derivations alpha-Derivations SAGBI Basis
publication/series
Master's Theses in Mathematical Sciences
report number
LUTFMA-3484-2022
ISSN
1404-6342
other publication id
2022:E42
language
English
id
9093857
2022-08-15 13:00:33
date last changed
2022-08-15 13:00:33
```@misc{9093857,
abstract     = {{In this text we continue the work of describing subalgebras of K[x] of ﬁnite codimension that was started in Describing subalgebras of K[x] using derivatives. In the referenced paper, the authors present how univariate subalgebras can be described by conditions based on evaluations in certain scalars, and proceed to develop a large theoretical framework to understand the nature of such conditions. The purpose of this thesis is to generalize as many of their results as possible to the multivariate setting K[x]. We include generalized deﬁnitions of the type, spectrum, clusters, α-derivations, and α, β-evaluation subtractions. We also state and prove generalizations of most of the theorems relating to the spectrum, clusters, α-derivation spaces, as well as the Main Theorem. We also give a couple of new results pertaining to how α-derivation spaces behave when we apply subalgebra conditions on clusters not containing α.}},
author       = {{Leffler, Erik}},
issn         = {{1404-6342}},
language     = {{eng}},
note         = {{Student Paper}},
series       = {{Master's Theses in Mathematical Sciences}},
title        = {{Describing Finite Codimensional Polynomial Subalgebras Using Partial Derivatives}},
year         = {{2022}},
}

```