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Full-Text Articles in Number Theory
Basis Criteria For N-Cycle Integer Splines, Ester Gjoni
Basis Criteria For N-Cycle Integer Splines, Ester Gjoni
Senior Projects Spring 2015
In this project we work with integer splines on graphs with positive integer edge labels. We focus on graphs that are n-cycles for some natural number n. We find an explicit condition for when a set of splines can form a module basis for n-cycle splines. In general, a set of splines forms a Z-module basis if and only if their determinant is equal to the product of the edge labels divided by the greatest common divisor of those edge labels.
Rational Tilings Of The Unit Square, Galen Dorpalen-Barry
Rational Tilings Of The Unit Square, Galen Dorpalen-Barry
Senior Projects Spring 2015
A rational n-tiling of the unit square is a collection of n triangles with rational side length whose union is the unit square and whose intersections are at most their boundary edges. It is known that there are no rational 2-tilings or 3-tilings of the unit square, and that there are rational 4- and 5-tilings. The nature of those tilings is the subject of current research. In this project we give a combinatorial basis for rational n-tilings and explore rational 6-tilings of the unit square.
Irreducibility And Galois Properties Of Lifts Of Supersingular Polynomials, Rylan Jacob Gajek-Leonard
Irreducibility And Galois Properties Of Lifts Of Supersingular Polynomials, Rylan Jacob Gajek-Leonard
Senior Projects Spring 2015
It has recently been shown that a rational specialization of Jacobi polynomials, when reduced modulo a prime number p, has roots which coincide with the supersingular j- invariants of elliptic curves in characteristic p. These supersingular lifts are conjectured to be irreducible with maximal Galois groups. Using the theory of p-adic Newton Polygons, we provide a new infinite class of irreducibility and, assuming a conjecture of Hardy and Littlewood, give strong evidence for their Galois groups being as large as possible.