Classification results for intersective polynomials with no integral roots

Abstract: In this talk, I describe the contents of my recently-defended PhD thesis on strongly intersective polynomials. These are polynomials with no integer roots but with a root modulo every positive integer, thereby constituting a failure of the local-global principle. We start by describing their relation to Hilbert's 10th Problem and an algorithm of James Ax. These are fascinating objects which make contact with many areas of math, including permutation group theory, splitting behaviour of prime ideals in number fields, and Frobenius elements from class field theory.

In particular, we discuss constraints on the splitting behaviour of ramified primes in splitting fields of intersective polynomials, building on the work of Berend-Bilu (1996) and Sonn (2008). We also explain the computation of a list of possible Galois groups of such polynomials, which includes many examples and which supports some recent conjectures of Ellis & Harper (2024).

Time permitting, we end by discussing future work, including results from permutation group theory and from character theory.

number theory

Audience: researchers in the topic

Carleton-Ottawa Number Theory seminar