#### Date of Graduation

7-2023

#### Document Type

Thesis

#### Degree Name

Bachelor of Science in Mathematics

#### Department

Mathematical Sciences

#### Advisor/Mentor

Mantero, Paolo

#### Committee Member/Reader

Harriss, Edmund

#### Committee Member/Second Reader

Zamboanga, Byron L.

#### Committee Member/Third Reader

Chapman, Kate

#### Abstract

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 the case in higher dimensions corresponding to polynomials in several variables. For this thesis, we will classify polynomials in three variables passing through a set of discrete points using an abstract algebraic structure known as an ideal. Then, we will analyze these ideals and specifically provide structural and numerical information. That is, we characterize their Hilbert Functions, which in our setting, are functions describing the number of linearly independent polynomials passing through the set of up to six points with a given multiplicity. Specifically, we will also see that in these cases, there is an expected ”maximal” Hilbert Function value, and the main goal is to determine whether these ideals have the ”maximal” Hilbert Function or not.

#### Keywords

Commutative Algebra, Hilbert Function, Algebraic Geometry

#### Citation

Xie, B.
(2023). Interpolation Problems and the Characterization of the Hilbert Function. * Mathematical Sciences Undergraduate Honors Theses*
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