Dissertations, Theses, and Capstone Projects

Date of Degree

6-2020

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Melvyn B. Nathanson

Committee Members

Kevin O'Bryant

Mark Sheingorn

Subject Categories

Number Theory

Keywords

Cantor Polynomials, Fueter-Polya Theorem, Lindemann-Weierstrass Theorem, Pairing Polynomials, Irrational Sectors

Abstract

A result by Fueter-Pólya states that the only quadratic polynomials that bijectively map the integral lattice points of the first quadrant onto the non-negative integers are the two Cantor polynomials. We study the more general case of bijective mappings of quadratic polynomials from the lattice points of sectors defined as the convex hull of two rays emanating from the origin, one of which falls along the x-axis, the other being defined by some vector. The sector is considered rational or irrational according to whether this vector can be written with rational coordinates or not. We show that the existence of a quadratic packing polynomial imposes restrictions on the sector, and we show that a non-zero discriminant will make injectivity of a quadratic polynomial impossible. A corollary is the non-existence of quadratic packing polynomials on irrational sectors. We then make a full classification of quadratic packing polynomials on rational sectors.

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