## Contact Info

Pure MathematicsUniversity of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

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∈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 ω_{1}(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_{i}q^{-}^{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

Event tags

University of Waterloo

200 University Avenue West

Waterloo, Ontario, Canada

N2L 3G1

Departmental office: MC 5304

Phone: 519 888 4567 x43484

Fax: 519 725 0160

Email: puremath@uwaterloo.ca

University of Waterloo

University of Waterloo

43.471468

-80.544205

200 University Avenue West

Waterloo,
ON,
Canada
N2L 3G1

The University of Waterloo acknowledges that much of our work takes place on the traditional territory of the Neutral, Anishinaabeg and Haudenosaunee peoples. Our main campus is situated on the Haldimand Tract, the land granted to the Six Nations that includes six miles on each side of the Grand River. Our active work toward reconciliation takes place across our campuses through research, learning, teaching, and community building, and is centralized within our Office of Indigenous Relations.