Number Theory Seminar: October 18, 2024
Location: 12:10 p.m. in Stevenson Center, room 1310
Speaker: Walter Bridges (University of North Texas)
Title: Zero attractors and sign changes in partition polynomials
Abstract: I will discuss new methods in the asymptotic theory of integer partitions with applications to the following two problems. The first problem concerns secondary terms in asymptotic equidistribution. For example, if $p(a,b,n)$ denotes the number of partitions of $n$ with parts congruent to $a$ modulo $b$, then it is easy to show that $p(a_1,b,n) \sim p(a_2,b,n)$ as $n \to \infty$ for all $0\leq a_1,a_2 < b$. On the other hand, the difference $p(a_1,b,n)-p(a_2,b,n)$ oscillates as $n \to \infty$ for any $a_1 \neq a_2$. A new technique allows us to predict the oscillation for this and similar problems.
The second problem concerns zero attractors for sequences of partition polynomials. If the coefficients of the polynomial $P_n(\zeta)$ count the number of partitions of $n$ into $m$ parts, then R. Stanley asked to identify the zero attractor of the sequence $P_n(\zeta)$ as $n \to \infty$ – that is, the set of limit points of the zero sets of the $P_n(\zeta)$. It was shown by Boyer and Goh that the zero attractor of $P_n(\zeta)$ is a “Pac-Man” shaped curve in the unit disk. We prove a zero attractor for partition polynomials that count hook lengths; somewhat surprisingly, the zero attractor features isolated points.
This is joint work with W. Craig, A. Folsom, J. Franke, T. Garnowski, J. Males and L. Rolen.