Project Overview
The polynomials for the Hilbert series of special monomial sets.
Department(s)
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