Quadratic patterns in irreducible polynomials over a finite field

Event time: 
Wednesday, February 10, 2021 - 4:15pm
Mark Shusterman
Speaker affiliation: 
Harvard University
Event description: 


Motivated by the centuries old conjecture that there are infinitely many integers n for which n^2+1 is a prime number, in a joint work with Will Sawin we show that over some finite field there are infinitely many polynomials f(x) for which the polynomial f(x)^2 + x is irreducible. Our proof combines analytic techniques previously used in the study of the aforementioned conjecture, with Grothendieck’s function-sheaf dictionary, and Deligne’s Riemann Hypothesis. We are then led to study the cohomology of local systems on the complement of an arrangement of hyperplanes in positive characteristic affine space, borrowing ideas from the analogous topological problem over the complex numbers.