The Least Common Multiple of a Bivariate Quadratic Sequence

TL;DR

This paper analyzes the asymptotic behavior of the least common multiple of polynomial values for quadratic polynomials in two variables, revealing four distinct growth patterns depending on the polynomial.

Contribution

It provides a complete classification of the asymptotic behaviors of the LCM of quadratic polynomial values, including deviations from a random model.

Findings

For generic polynomials, the LCM growth is about N log log N / sqrt(log N).

Special polynomials exhibit different asymptotic behaviors, some deviating from the random model.

The paper characterizes when each asymptotic behavior occurs for quadratic polynomials.

Abstract

Let $F\in\mathbb{Z}[x,y]$ be some polynomial of degree 2. In this paper we find the asymptotic behaviour of the least common multiple of the values of $F$ up to $N$. More precisely, we consider $\psi_F(N) = \log\left(\text{LCM}_{0<F(x,y)\leq N}\left\lbrace F(x,y)\right\rbrace\right)$ as $N$ tends to infinity. It turns out that there are 4 different possible asymptotic behaviours depending on $F$. For a generic $F$, we show that the function $\psi_F(N)$ has order of magnitude $\frac{N\log\log N}{\sqrt{\log N}}$. We also show that this is the expected order of magnitude according to a suitable random model. However, special polynomials $F$ can have different behaviours, which sometimes deviate from the random model. We give a complete description of the order of magnitude of these possible behaviours, and when each one occurs.