On Artin's Conjecture: Pairs of Additive Forms

Abstract: A conjecture by Emil Artin claims that for forms $f_1, \dots, f_r \in \mathbb{Z}[x_1, \dots, x_s]$ of degree $k_1, \dots, k_r$ the system of equation $f_1=f_2=\dots=f_r=0$ has a non-trivial $p$-adic solution for all primes $p$ provided that $s > k_1^2 + \dots + k_r^2$. Although this conjecture was disproved in general, it holds in some cases. In this talk I will focus on the case of two additive forms with the same degree $k$ and sketch the proof that Artin's conjecture holds in this case unless $k=2^\tau$ for $2 \le \tau \le 15$ and $k=3\cdot 2^\tau$ for $2 \le \tau$.

algebraic geometrycombinatoricsdynamical systemsgeneral topologynumber theory

Audience: researchers in the topic

ZORP (zoom on rational points)

Series comments: 2 talks on a Friday, roughly once per month.

Online coffee break in between.