Bounded generation of SL2 over rings of S-integers with infinitely many units

Gl-equivariant Modules over Polynomial Rings in Infinitely Many Variables

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, sys...

Group Rings over the p-Adic Integers

Hilbert’s Tenth Problem over Rings of Integers of Number Fields

This is a term paper for Thomas Scanlon’s Math 229, Model Theory, at UC Berkeley, Fall 2006. The goal of this survey is to understand Bjorn Poonen’s theorem about elliptic curves and Hilbert’s Tenth problem over rings of integers of number fields and to record and exposit a few ideas which may be useful in an attack with Poonen’s theorem as a starting point.

Polynomial Functions over Rings of Residue Classes of Integers

In this thesis we discuss how to find equivalent representations of polynomial functions over the ring of integers modulo a power of a prime. Specifically, we look for lower degree representations and representations with fewer variables for which important applications in electrical and computer engineering exist. We present several algorithms for finding these compact formulations. INDEX WORD...

Infinitely Many Periodic Solutions of Nonlinear Wave Equations on S

The existence of time periodic solutions of nonlinear wave equations utt −∆nu + ` n− 1 2 ́2 u = g(u)− f(t, x) on n-dimensional spheres is considered. The corresponding functional of the equation is studied by the convexity in suitable subspaces, minimax arguments for almost symmetric functional, comparison principles and Morse theory. The existence of infinitely many time periodic solutions is o...