Algebra Commons^™

Algebraic And Integral Closure Of A Polynomial Ring In Its Power Series Ring, Joseph Swanson

All Dissertations

Let R be a domain. We look at the algebraic and integral closure of a polynomial ring, R[x], in its power series ring, R[[x]]. A power series α(x) ∈ R[[x]] is said to be an algebraic power series if there exists F (x, y) ∈ R[x][y] such that F (x, α(x)) = 0, where F (x, y) ̸ = 0. If F (x, y) is monic, then α(x) is said to be an integral power series. We characterize the units of algebraic and integral power series. We show that the only algebraic power series with infinite radii of convergence are …

Interpolation Problems And The Characterization Of The Hilbert Function, Bryant Xie

Mathematical Sciences Undergraduate Honors Theses

In mathematics, it is often useful to approximate the values of functions that are either too awkward and difficult to evaluate or not readily differentiable or integrable. To approximate its values, we attempt to replace such functions with more well-behaving examples such as polynomials or trigonometric functions. Over the algebraically closed field C, a polynomial passing through r distinct points with multiplicities m1, ..., mr on the affine complex line in one variable is determined by its zeros and the vanishing conditions up to its mi − 1 derivative for each point. A natural question would then be to consider …

Families Of Homogeneous Licci Ideals, Jesse Keyton

Graduate Theses and Dissertations

This thesis is concered with the graded structure of homogeneous CI-liaison. Given two homogeneous ideals in the same linkage class, we want to understand the ways in which you can link from one ideal to the other. We also use homogeneous linkage to study the socles and Hilbert functions of Artinian monomial ideals.

First, we build off the work of C. Huneke and B. Ulrich on monomial liaison. They provided an algorithm to check the licci property of Artinian monomial ideals and we use their method to characterize when two Artinian monomial ideals can be linked by monomial regular sequences. …

Gröbner Bases And Systems Of Polynomial Equations, Rachel Holmes

All Graduate Theses, Dissertations, and Other Capstone Projects

This paper will explore the use and construction of Gröbner bases through Buchberger's algorithm. Specifically, applications of such bases for solving systems of polynomial equations will be discussed. Furthermore, we relate many concepts in commutative algebra to ideas in computational algebraic geometry.

Geometry Of Linear Subspace Arrangements With Connections To Matroid Theory, William Trok

Theses and Dissertations--Mathematics

This dissertation is devoted to the study of the geometric properties of subspace configurations, with an emphasis on configurations of points. One distinguishing feature is the widespread use of techniques from Matroid Theory and Combinatorial Optimization. In part we generalize a theorem of Edmond's about partitions of matroids in independent subsets. We then apply this to establish a conjectured bound on the Castelnuovo-Mumford regularity of a set of fat points.

We then study how the dimension of an ideal of point changes when intersected with a generic fat subspace. In particular we introduce the concept of a ``very unexpected hypersurface'' …

On Rings Of Invariants For Cyclic P-Groups, Daniel Juda

Graduate Theses and Dissertations

This thesis studies the ring of invariants R^G of a cyclic p-group G acting on k[x_1,\ldots, x_n] where k is a field of characteristic p >0. We consider when R^G is Cohen-Macaulay and give an explicit computation of the depth of R^G. Using representation theory and a result of Nakajima, we demonstrate that R^G is a unique factorization domain and consequently quasi-Gorenstein. We answer the question of when R^G is F-rational and when R^G is F-regular.

We also study the a-invariant for a graded ring S, that is, the maximal graded degree of the top local cohomology module of S. …

Multiplicative Sets Of Atoms, Ashley Nicole Rand

Doctoral Dissertations

It is possible for an element to have both an atom factorization and a factorization that will always contain a reducible element. This leads us to consider the multiplicatively closed set generated by the atoms and units of an integral domain. We start by showing that for a nice subset S of the atoms of R, there exists an integral domain containing R with set of atoms S. A multiplicatively closed set is saturated if the factors of each element in the set are also elements in the set. Considering polynomial and power series subrings, we find necessary and sufficient …

Algebraic And Combinatorial Properties Of Certain Toric Ideals In Theory And Applications, Sonja Petrovic

University of Kentucky Doctoral Dissertations

This work focuses on commutative algebra, its combinatorial and computational aspects, and its interactions with statistics. The main objects of interest are projective varieties in Pⁿ, algebraic properties of their coordinate rings, and the combinatorial invariants, such as Hilbert series and Gröbner fans, of their defining ideals. Specifically, the ideals in this work are all toric ideals, and they come in three flavors: they are defining ideals of a family of classical varieties called rational normal scrolls, cut ideals that can be associated to a graph, and phylogenetic ideals arising in a new and increasingly popular area of …