# Project Overview

## The polynomials for the Hilbert series of special monomial sets.

Gabe Sosa Castillo (gsosacastillo@colgate.edu)

Mathematics

### Abstract

Given a polynomial ring R and an ideal generated by a set of homogeneous polynomials (or monomials), I, one can define the Hilbert function as a function associating a natural number n to the number of polynomials of degree n that are not in the initial ideal of I, i.e. that are not divisible by the generating monoimials of the initial ideal of I.

As an example if I is the ideal generated by xy, yz and xz, then f(n)=1 for n=0 and f(n)=3 for n>0, because the monomials of degree n that are not divisible by xy, yz, or xz, are the nth powers of x, y and z.

The Hilbert series of the quotient ring R/I is the series f(0)+f(1)t+f(2)t^2+f(3)t^3+... is a power series that can always be written as P(t)/(1-t)^d for some d in the natural numbers, P(t) is the polynomial of the HIlbert series of R/I. P(t) encodes algebraic invariants about the ring R/I.

This project will focus on determining P(t) and/or bounds for its degree, when the ideal I is either associated to a graph G or I satisfies strongly stable conditions. This research will involve interplay between algebra and combinatorics, i.e. understanding how invariants of a graph such as independence number, matching number, number and lengths of cycles, affect the degree of the polynomial for the Hilbert series. Experimental computations will help produce data.

### Student Qualifications

Required: MATH 214 Linear Algebra and experience with writing mathematical proofs as evidenced by a grade of B, or higher, in a proof based course in Mathematics (for example MATH 250, number theory and mathematical reasoning, MATH 310, Combinatorial Problem Solving, MATH 360, Graph Theory, or MATH 377, Real Analysis I).

Preferred: Programming experience, successfully completing MATH 375 (Abstract Algebra I).

### Number of Student Researchers

Two (without including the STAR scholar) students

### Project Length

Eight weeks

Applications open on 10/03/2023 and close on 02/28/2024