Week of November 22, 2020

Sun Nov 22, 2020
1:21pm - Zoom: https://uci.zoom.us/j/95483608618 - Number Theory
Andrei Mandelshtam - (UC Irvine)
The Structure of the Positive Monoid of Integer-Valued Polynomials Evaluated at an Algebraic Number
In the ring $\mathbb{Q}[x]$ of polynomials with coefficients in the rational numbers, it is interesting to consider the subring of all integer-valued polynomials, i.e. polynomial $p(x)$ such that $p(n)$ is an integer for every integer $n$. This ring is known as the most natural and simple example of a non-Noetherian ring. One may wonder whether this is not just the set of all polynomials with integer coefficients. However, e.g. the polynomial $(x^2+x)/2$ is integer-valued. It turns out that this ring consists of exactly the polynomials with integer coefficients in the basis of binomial coefficients $\binom{x}{n}$. Motivated by the characterization of symmetric monoidal functors between Deligne categories, we examine the set $R_{+}(x)$ of polynomials which have nonnegative integer coefficients in this basis. More precisely, we study the set of values of these polynomials at a fixed number $\alpha$. It turns out that this set has a fascinating algebraic structure, explicitly determined by the $p$-adic roots of the minimal polynomial of $\alpha$, which we will fully describe in this talk. This work is joint with Daniil Kalinov, MIT.