# Student Number Theory Seminar

Tuesday, May 30, 2023 — 9:30 AM EDT

Talk #1: Sourabhashis Das, Department of Pure Mathematics, University of Waterloo

"On the number of irreducible factors with a given multiplicity in function fields"

Let k≥1 be a natural number and f∈Fq[t] be a monic polynomial. Let ωk(f) denote the number of distinct monic irreducible factors of f with multiplicity k. In this talk, we show that the function ω1(f) has a normal order log(deg(f)) and also satisfies the Erdös-Kac Theorem. We also show that the functions ωk(f) with k≥2 do not have normal order. Such results are obtained by studying the first and the second moments of ωk(f) which we explain in brief. This is joint work with Ertan Elma, Wentang Kuo, and Yu-Ru Liu.

Talk #2: Liam Orovec, Department of Pure Mathematics, University of Waterloo

"Small Univoque Bases"

For a positive number q, we say (εi) is a q-expansion for x provided, x=∑εiq-i. Working over the alphabet A={0,1,…,M} we look at finding, given a fixed positive real number x, the smallest base qs(x) for which x has a unique qs(x)-expansion.

We will first establish the result for x=1. Then using relations between the representation of 1 under base qs(x) and the possible unique representation of real numbers we determine whether qs(x)≤qs(1) which will aid us in calculating the desired value.

This is a generalization of the work of D. Kong who established the results for M=1. The study of such bases is important, as most x have an infinite number of representations under an arbitrary base q.

MC 5417

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