Date of Award
9-1-2020
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
Earnest, Andrew
Second Advisor
Kocik, Jerzy
Abstract
An n-dimensional integral quadratic form over Z is a polynomial of the form f = f(x1, … ,xn) =∑_(1≤i,j ≤n)▒a_ij x_i x_j, where a_ij=a_ji in Z. An integral quadratic form is called positive definite if f(α_1, …,α_n) > 0 whenever (0, … , 0) ≠(α_1, …,α_n) in Z^n. A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than almost universality. Main results of this study are: every primitively universal form non-trivially represents zero over every ring Z_p of p-adic integers, and every almost universal form in five or more variables is almost primitively universal. With use of these results and improving a result of G. Pall from 1946, we then provide criteria to determine whether a given integral quadratic lattice over a ring Z_p of p-adic integers is Z_p-universal or primitively Z_p-universal. The criteria are stated explicitly in terms of a Jordan splitting of the lattice. As an application of the local criteria, we complete the determination of the universal positive definite classically integral quaternary quadratic forms that are almost primitively universal, which was initiated in work of N. Budarina in 2010. Finally, with the use of these local results, we identify 28 positive definite classically integral primitively universal quaternary quadratic forms which were not known previously, introducing a conjecture obtained by a numerical approach, which could possibly be the primitive counterpart to the Fifteen Theorem proved by J.H. Conway and W.A. Schneeberger in 1993.
Access
This dissertation is Open Access and may be downloaded by anyone.