Department

Mathematics Department

First Advisor

Dr. Tyrrell McAllister

Description

A very important problem in discrete geometry is counting points with integer-coordinates (also called “lattice-points”) in polytopes. A polytope is a geometric shape that is the smallest convex set containing the vertices defining the polytope. For instance, in two dimensions, a polytope would simply be a convex polygon. Lattice-point enumeration has applications in a lot of different areas of mathematics, including combinatorics and operations research. Our goal is to study the function that counts the lattice-points in a polytope and its integer dilates when the vertices of the polytope have rational coordinates. This function is a quasi-polynomial. That is to say, it is a polynomial with coefficients that are themselves periodic functions in the variable, meaning that the values of those functions repeat themselves after a certain period. This function is the Ehrhart quasi-polynomial of the polytope. For this research, I am focusing on the periods of the coefficients of the Ehrhart quasi-polynomial in order to see if they can take on any value for some polytope or if there exist restrictions on these periods. It has already been proven that no interesting restrictions exist for the 2-dimensional case and we have constructed a family of polytopes that proves that no interesting restrictions exist for the non-convex 3-dimensional case. We are also able to generalize some of our contructions to higher dimensions.

Comments

EPSCoR, Honors Program

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Lattice-points enumeration in polytopes Study of the coefficients of the Ehrhart quasi-polynomial

A very important problem in discrete geometry is counting points with integer-coordinates (also called “lattice-points”) in polytopes. A polytope is a geometric shape that is the smallest convex set containing the vertices defining the polytope. For instance, in two dimensions, a polytope would simply be a convex polygon. Lattice-point enumeration has applications in a lot of different areas of mathematics, including combinatorics and operations research. Our goal is to study the function that counts the lattice-points in a polytope and its integer dilates when the vertices of the polytope have rational coordinates. This function is a quasi-polynomial. That is to say, it is a polynomial with coefficients that are themselves periodic functions in the variable, meaning that the values of those functions repeat themselves after a certain period. This function is the Ehrhart quasi-polynomial of the polytope. For this research, I am focusing on the periods of the coefficients of the Ehrhart quasi-polynomial in order to see if they can take on any value for some polytope or if there exist restrictions on these periods. It has already been proven that no interesting restrictions exist for the 2-dimensional case and we have constructed a family of polytopes that proves that no interesting restrictions exist for the non-convex 3-dimensional case. We are also able to generalize some of our contructions to higher dimensions.