Number Theory Seminar
Pete Clark
University of Georgia
A Group-Theoretic Ax-Katz Theorem
Abstract: The Chevalley-Warning Theorem says that for a system of polynomials of
"low degree"(the sum of the degrees is less than the number of variables)
over a finite field of characteristic p, the number of solutions is a
multiple of p. The Ax-Katz Theorem refines this to the best p-adic
divisibility on the size of the solution set in terms of the degrees of
the polynomials and the number of variables. Recently Aichinger-Moosbauer
developed a calculus of finite differences for maps between commutative
groups and used this to generalize Chevalley-Warning to any finite
rng of prime power order. Remarkably, they give a purely group-theoretic
result from which their ring-theoretic result follows. I will present
an extension of the Ax-Katz Theorem to finite rings of characteristic p,
which again follows from a purely group-theoretic result stated in terms
of the Aichinger-Moosbauer calculus. This is joint with U. Schauz. If
time permits, I will mention work towards extending this result to all
finite p-groups, joint with Schauz and N. Triantafillou.
Zoom link:
https://uic.zoom.us/j/88173268700?pwd=aEhmTGpSOVhidWE4L1VWUnNhNVlvUT09
Friday April 1, 2022 at 1:00 PM in Zoom