A universal Hilbert set is an infinite set S ⊆ Z having the property that for every F(x,y) ∈ Z[x,y] which is irreducible in Q[x, y] and satisfies degx(F ) ≥ 1, we have that for all but finitely many y0 ∈ S, the polynomial F(x,y0) is irreducible in 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 ≥ 1, and explain how a result of K. Ford (2008) implies the existence of a universal Hilbert set S satisfying |{m ∈ Z : m ̸∈ S,|m| ≤ X}| ≪ X /(log X)^δ where δ = 1−(1+loglog2)/(log2) = 0.086071…. This is joint work with Robert Wilcox.

Upstate NY online number theory colloquium

See for zoom link and password:

http://people.math.binghamton.edu/borisov/UpstateNYOnline/Colloquium.html