Towards Artin’s conjecture on $p$-adic forms in low degree

Abstract: Artin conjectured that whenever $n$ is at least $d^2$, a homogeneous polynomial $f(x_0, ..., x_n)$ of degree $d$ in $n+1$ variables has a nontrivial $p$-adic zero; equivalently, the field $\mathbb{Q}_p$ is $C_2$. This conjecture is false, with the first counterexample in degree 4 over $\mathbb{Q}_2$ discovered by Terjanian. However, all known counterexamples have composite degree, begging the question: does Artin's conjecture hold if we restrict to prime degrees $d$? We have evidence in degrees 2 and 3 due to results of Hasse and Lewis, respectively, and the celebrated Ax--Kochen theorem establishes an asymptotic version of Artin's conjecture when $p$ is large relative to $d$. In recent joint work with Lea Beneish, we establish Artin's conjecture in degree 5 when $p > 5$ and in degree 7 when $p > 679$. In this talk, we will explore the ideas and techniques spanning nearly a century behind these results, from the Lang--Weil theorem to effective Bertini theorems and parallel computing.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.