Algebra, Geometry, and Number Theory Seminar

Abstracts

Nathan McNew - Covering numbers and integer programming in combinatorial number theory

A positive integer n is a covering number if there exists a distinct covering system that uses only divisors of n as moduli. These numbers form a strict subset of the abundant numbers, whose density has been studied and refined by various authors over the last century. We establish that the set of covering numbers C have a natural density and prove 0.10323 < d(C) < 0.103398. Determining whether a given integer is a covering number is computationally difficult, so we develop a method, using integer programming to determine whether integers are covering numbers, which allows us to extend the range of known covering numbers much further than previous calculations. Using the Gurobi solver, we classify all integers up to 773,500, identifying all primitive covering numbers in this range, establishing the lower bound for d(C), and also identifying new families of primitive covering numbers. For the upper bounds we also introduce new methods to speed up the calculation. These same methods also allow us to improve the bounds for abundant numbers. Time permitting, we will discuss broader applications of these integer programming techniques to other problems in combinatorial number theory, including determining subsets of the integers avoiding three-term geometric progressions, where IP formulations improve similar density computations and allude to a deeper structure that is still not well understood.

Michael Filaseta - Excursions into the factorization of lacunary polynomials

By a lacunary (or sparse) polynomial, we loosely mean a polynomial of high degree with few terms. The main interest for the talk is in polynomials in one variable over the rationals. We will begin with some interesting history, starting with a still open problem of Turàn that basically asserts that every polynomial in $\mathbb Z[x]$ is "close" (to be clarified) to an irreducible polynomial. The history will lead us to the main part of the talk which will be on the factorization of polynomials of the general form \[ f_{0}(x) + f_{1}(x) x^{n} + \cdots + f_{r}(x) x^{rn} \in \mathbb Z[x], \] with \[ r \ge 1, \quad \gcd{}_{\mathbb Z[x]}\big(f_{0}(x), f_{1}(x), \ldots, f_{r}(x) \big) = 1, \quad f_{0}(0) \ne 0, \quad \text{and} \quad f_{r}(x) \ne 0. \] We will describe a method which typically allows us to say that for $n$ large (depending on $r$ and the $f_{j}(x)$), such polynomials are a product (possibly empty) of cyclotomic polynomials and an irreducible polynomial. The talk will focus on the methods and illustrative examples rather than proofs.As time permits, other results, of a different nature, on the factorization of lacunary polynomials will be discussed.

Arindam Roy - TBA

Last semester's seminar.