Towards Artin's conjecture on $p$-adic quintic forms

Abstract: Let $K/\mathbb{Q}_p$ be a finite extension with residue field $\mathbb{F}_q$ and suppose $f(x_0, \ldots, x_n)$ is a homogeneous polynomial of degree $d$ over $K$. A conjecture, originally due to Artin, states that when $d$ is prime and $n \geq d^2$, $f=0$ has a nontrivial solution in $K$. This conjecture is known in degrees 2 and 3 due to Hasse and Lewis, respectively. It is also "asymptotically true," due to work of Ax and Kochen, in that it holds when $q$ is sufficiently large with respect to $d$, though this is difficult to make effective. In this talk, we present recent joint work with Lea Beneish in which we prove the quintic version of the conjecture holds if $q \geq 7$. Our methods include both a refinement to a geometric approach of Leep and Yeomans (who showed $q \geq 47$ suffices) and a significant computational component.

number theory

Audience: researchers in the topic

London number theory seminar

Series comments: For reminders, join the (very low traffic) mailing list at mailman.ic.ac.uk/mailman/listinfo/london-number-theory-seminar

For a record of talks predating this website see: wwwf.imperial.ac.uk/~buzzard/LNTS/numbtheo_past.html