#### Date of Graduation

8-2018

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics (PhD)

#### Degree Level

Graduate

#### Department

Mathematical Sciences

#### Advisor

Mark Johnson

#### Committee Member

Lance Miller

#### Second Committee Member

Wenbo Niu

#### Keywords

Closures, Commutative Algebra, Hilbert-Kunz, Hilbert-Samuel, Multiplicity

#### Abstract

We define a family of functions, called s-multiplicity for each s>0, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an associativity formula generalizing the associativity formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures, called s-closures, such that if two ideals have the same s-closure then they have the same s-multiplicity, and the converse holds under mild conditions. We describe methods for computing the $F$-threshold, the $s$-multiplicity, and the $s$-closure of monomial ideals in toric rings using the geometry of the cone defining the ring.

#### Recommended Citation

Taylor, William D., "Interpolating Between Multiplicities and F-thresholds" (2018). *Theses and Dissertations*. 2842.

http://scholarworks.uark.edu/etd/2842