Student Number Theory Seminar

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∈F_q[t] be a monic polynomial. Let ω_k(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 ω_k(f) with k≥2 do not have normal order. Such results are obtained by studying the first and the second moments of ω_k(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 (ε_i) is a q-expansion for x provided, x=∑ε_iq^-i. Working over the alphabet A={0,1,…,M} we look at finding, given a fixed positive real number x, the smallest base q_s(x) for which x has a unique q_s(x)-expansion.

We will first establish the result for x=1. Then using relations between the representation of 1 under base q_s(x) and the possible unique representation of real numbers we determine whether q_s(x)≤q_s(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