NTAG Seminar: Towards Artin’s conjecture on p-adic forms in low degree

Artin conjectured that whenever n is at least d^2, a homogeneous polynomial f(x0, ..., xn) of degree d in n+1 variables has a nontrivial p-adic zero; equivalently, the field Qp is C2. This conjecture is false, with the first counterexample discovered by Terjanian in degree 4 over Q2. 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 counting points on varieties over finite fields to effective Bertini theorems and parallel computing.