On a dense universal Hilbert set

Abstract: A $universal\ Hilbert\ set$ is an infinite set $\mathcal S \subseteq \mathbb Z$ having the property that for every $F(x,y) \in \mathbb Z[x,y]$ which is irreducible in $\mathbb Q[x,y]$ and satisfies $\deg_{x} (F) \ge 1$, we have that for all but finitely many $y_{0} \in \mathcal S$, the polynomial $F(x,y_{0})$ is irreducible in $\mathbb Q[x]$. The existence of universal Hilbert sets is due to P.C. Gilmore and A. Robinson in 1955, and since then a number of explicit examples have been given. Universal Hilbert sets of density $1$ in the integers have been shown to exist by Y. Bilu in 1996 and P. D\`ebes and U. Zannier in 1998. In this talk, we discuss a connection between universal Hilbert sets and Siegel's Lemma on the finiteness of integral points on a curve of genus $\ge 1$, and explain how a result of K.Ford (2008) implies the existence of a universal Hilbert set $\mathcal S$ satisfying \[ |\{ m \in \mathbb Z: m \not\in \mathcal S, |m| \le X \}| \ll \dfrac{X}{(\log X)^{\delta}}, \] where $\delta = 1 - (1+\log\log 2)/(\log 2) = 0.086071\ldots$. This is joint work with Robert Wilcox.

number theory

Audience: researchers in the discipline

Upstate New York Online Number Theory Colloquium