Abstract

What does a “typical” polynomial look like? Suppose we build a polynomial of degree n by choosing each coefficient independently and at random to be either (+1) or (-1). A natural question then arises: how likely is such a polynomial to be irreducible over the integers?

Despite how simple this question sounds, it remains largely open and has attracted a great deal of attention from mathematicians. Over the years, researchers have discovered that understanding random polynomials requires ideas from many different areas of mathematics.

In this talk, we will take a journey through the mathematics behind this problem. We will start with classical results about polynomial irreducibility dating back to the 19th century, and then move toward modern perspectives on random polynomials. Along the way, we will see how tools from different fields come together to tackle this deceptively simple question, ending with some exciting developments that appeared as recently as last few years.

No prior background in probability or advanced algebra will be assumed.