Number Theory Seminar

Thursday, February 28, 2019
4:00pm to 5:00pm
On the Number of Monic Polynomials over $\mathbb{F}_q$ of a Given Discriminant and Degree
For a fixed integer $m>1$ and finite field $F$ of odd order $q$, we find an exact formula for the number of degree-$m$ monic polynomials over $F$ of a given discriminant $d$. As one application, we prove that the discriminant is equally distributed among such polynomials if and only if $gcd(q-1,m(m-1))=2$. As another application, we explicitly compute the global Hasse-Weil zeta function of hypersurfaces cut out by the discriminant function, and express them in terms of Hecke characters of cyclotomic fields. In particular, we verify new cases of the Hasse-Weil conjecture for an infinite family of varieties of unbounded dimension.