Oliver Club

Anton MosunovUniversity of Waterloo
Representations of Integers by Binary Forms

Thursday, February 15, 2024 - 4:00pm
Malott 532 (Lounge)

A homogeneous polynomial in two variables is called a binary form. The non-zero terms of such a homogeneous polynomial all have the same degree, and we refer to it as the degree of a binary form. For example, $x^3 - 7xy^2 + 5y^3$ is a binary form of degree $3$, while $6x^5y - 20x^3y^3 + 6xy^5$ is a binary form of degree $6$.

Now, let $F(x, y)$ be a binary form with integer coefficients of positive degree. We say that an integer $m$ is \emph{representable} by $F$ if and only if there exist integers $x$ and $y$ such that $F(x, y) = m$. Let $R_F(N)$ denote the number of integers $m$ of absolute value at most $N$ that are representable by $F$.\linebreak In this talk we will discuss what is known about the asymptotic growth rate of the function $R_F(N)$ as $N$ tends to infinity. The main focus of our discussion will be on the case when the degree of $F$ exceeds $2$ and related problems, such as estimating the area of a so-called \emph{fundamental domain} $\{(x, y) \in \mathbb R^2 \colon |F(x, y)| \leq 1\}$ of $F$.

Refreshments will be served at 3:30 PM.