Mark Shusterman (Tel-Aviv University)
Irreducibility of integral polynomials with a large gap

Providing irreducibility criteria for integral polynomials is by now a classical topic. Yet, the irreducibility of "most" polynomials cannot be established using the existing techniques, and many problems remain open.

For example, establishing the irreducibility of random polynomials, and the irreducibility of various trinomials.
We will be interested in polynomials with only few nonzero coefficients, located "near the ends" (that is, a large gap in the middle). Focusing on a particular case, the family of polynomials X^{2k+1} -7X^2 + 1, we show how work of Schinzel (and generalizations by Bombieri-Zannier using unlikely intersections) imply irreducibility for large enough  k. We then discuss work by Filaseta-Ford-Konyagin which makes this reasonably effective.

This is a joint work with Will Sawin and Michael Stoll.