Publication Date

Spring 5-5-2025

Document Type

Thesis

Degree Name

Master of Science in Mathematics (MS)

Department

Mathematics

Committee Chair

Rohitha Goonatilake

Committee Member

Norma Saikali

Committee Member

Runchang Lin

Committee Member

Nilda M. Garcia

Abstract

In 1890, David Hilbert published a set of notes on what now constitutes one of the bases of Commutative Algebra; his work would eventually influence the efforts of mathematicians like Ernst Kunz. In 1969, Ernst Kunz introduced a particular mapping regarding modules of regular local rings. His goal was to characterize Noetherian local rings of prime characteristic by computing the length of the composition series under Frobenius power transformations. In this thesis, the focus will be on stating the initial steps on finding the coefficients of the Hilbert-Kunz function of the normal affine semigroup ring of the form R = k[u, su, s^2 u, ..., s^a u, lu, slu, s^2 lu, ..., s^b l^h u] with characteristic p = 2 which is associated with the affine semigroup ring generated by A(0, 0, 1), B(0, a, 1), C(2a, a, 1), and D(a, 0, 1) in Z^3 . The Hilbert-Kunz function will determine the length of the module of such a Noetherian ring when under frobenious power transformations. Finding such length offers a measure on the severity of the singularity of the ring as well as growth behavior insights. The initial geometric approach will require the use of Pick’s theorem, Ehrhart theory, and software like Macaulay2 and GeoGebra to compute the number of lattice points enclosed by the convex hull that corresponds to aforementioned normal affine semigroup ring.

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