Zero Spectrum Subalgebras of K[x] Described by Higher Derivatives
(2023) In Bachelor's Theses in Mathematical Sciences FMAL01 20232Mathematics (Faculty of Engineering)
- Abstract
- Unital subalgebras of finite codimension in the polynomial ring $\mathbb{K}[x]$ are described by a finite number of so called subalgebra conditions over a finite set in $\mathbb{K}$ named the subalgebra spectrum. Restricting attention to subalgebras whose spectrum is the singleton $\{0\}$ reveals a rather well behaved class of subalgebras, called almost monomial from the fact that these contain an ideal consisting of all monomials above a certain degree. Analysing them is made smoother with the introduction of the lower degree of a polynomial, giving rise to a lower numerical semigroup and, in turn, a linear basis consisting of a finite number of basis vectors from a quotient along with all monomials of sufficiently large degrees. The main... (More)
- Unital subalgebras of finite codimension in the polynomial ring $\mathbb{K}[x]$ are described by a finite number of so called subalgebra conditions over a finite set in $\mathbb{K}$ named the subalgebra spectrum. Restricting attention to subalgebras whose spectrum is the singleton $\{0\}$ reveals a rather well behaved class of subalgebras, called almost monomial from the fact that these contain an ideal consisting of all monomials above a certain degree. Analysing them is made smoother with the introduction of the lower degree of a polynomial, giving rise to a lower numerical semigroup and, in turn, a linear basis consisting of a finite number of basis vectors from a quotient along with all monomials of sufficiently large degrees. The main result of this thesis is that the subalgebra conditions of almost monomial algebras are found from the annihilator of this quotient. Hence the subalgebra conditions are found from solving a matrix kernel problem, given a linear basis of this particular kind. By applying this result to one of the simplest kind of almost monomial subalgebras, a proof for the existence of a previously conjectured sequence of derivations is revealed. (Less)
- Popular Abstract
- In the same sense that numbers can be added, subtracted, and multiplied, this too can be done on expressions for mathematical equations. For example $2x^2+x$ is the addition of $2x^2$ and $x$, and $2x^2$ itself may be seen as the multiplication of $2x$ and $x$. If instead we constrict which equations we consider, multiplying two such equations may result in an entirely different type of equation. As an example, all quadratic equations can be added and still end up as a quadratic equation. However, multiplying two quadratic equations will result in a fourth order equation, as for example $(x^2-1)\cdot x^2 = x^4-x^2$. In a sense, multiplication is not a permitted calculation among quadratic equations. If it is possible to find a collection... (More)
- In the same sense that numbers can be added, subtracted, and multiplied, this too can be done on expressions for mathematical equations. For example $2x^2+x$ is the addition of $2x^2$ and $x$, and $2x^2$ itself may be seen as the multiplication of $2x$ and $x$. If instead we constrict which equations we consider, multiplying two such equations may result in an entirely different type of equation. As an example, all quadratic equations can be added and still end up as a quadratic equation. However, multiplying two quadratic equations will result in a fourth order equation, as for example $(x^2-1)\cdot x^2 = x^4-x^2$. In a sense, multiplication is not a permitted calculation among quadratic equations. If it is possible to find a collection of equations where multiplication is always allowed, we call this an algebra.
In most algebras, one can form a checklist of conditions which an equation must satisfy to be part of the algebra. These requirements are often difficult to find. In this thesis, we define and explore a special kind of algebra named almost monomial, where we give an easy way to find the list of conditions which need to be satisfied. In particular, to find the checklist one needs to solve a linear system of equations. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9142164
- author
- Kennerland, Erik LU
- supervisor
- organization
- course
- FMAL01 20232
- year
- 2023
- type
- M2 - Bachelor Degree
- subject
- keywords
- Algebra, Almost Monomial Subalgebra, Derivation, Lower Degree, Lower Semigroup, Subalgebra Spectrum, Truncation
- publication/series
- Bachelor's Theses in Mathematical Sciences
- report number
- LUTFMA-4012-2023
- ISSN
- 1654-6229
- other publication id
- 2023:K28
- language
- English
- id
- 9142164
- date added to LUP
- 2024-02-12 15:46:26
- date last changed
- 2024-02-12 15:46:26
@misc{9142164,
abstract = {{Unital subalgebras of finite codimension in the polynomial ring $\mathbb{K}[x]$ are described by a finite number of so called subalgebra conditions over a finite set in $\mathbb{K}$ named the subalgebra spectrum. Restricting attention to subalgebras whose spectrum is the singleton $\{0\}$ reveals a rather well behaved class of subalgebras, called almost monomial from the fact that these contain an ideal consisting of all monomials above a certain degree. Analysing them is made smoother with the introduction of the lower degree of a polynomial, giving rise to a lower numerical semigroup and, in turn, a linear basis consisting of a finite number of basis vectors from a quotient along with all monomials of sufficiently large degrees. The main result of this thesis is that the subalgebra conditions of almost monomial algebras are found from the annihilator of this quotient. Hence the subalgebra conditions are found from solving a matrix kernel problem, given a linear basis of this particular kind. By applying this result to one of the simplest kind of almost monomial subalgebras, a proof for the existence of a previously conjectured sequence of derivations is revealed.}},
author = {{Kennerland, Erik}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor's Theses in Mathematical Sciences}},
title = {{Zero Spectrum Subalgebras of K[x] Described by Higher Derivatives}},
year = {{2023}},
}