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SUMMARY:Chris Keyes (KCL)
DTSTART:20251014T130000Z
DTEND:20251014T140000Z
DTSTAMP:20260421T154239Z
UID:UEAPS/59
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/UEAPS/59/">T
 owards Artin’s conjecture on $p$-adic forms in low degree</a>\nby Chris 
 Keyes (KCL) as part of ANTLR seminar\n\n\nAbstract\nArtin 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\; eq
 uivalently\, the field $\\mathbb{Q}_p$ is $C_2$. This conjecture is false\
 , with the first counterexample in degree 4 over $\\mathbb{Q}_2$ discovere
 d by Terjanian. However\, all known counterexamples have composite degree\
 , begging the question: does Artin's conjecture hold if we restrict to pri
 me degrees $d$? We have evidence in degrees 2 and 3 due to results of Hass
 e and Lewis\, respectively\, and the celebrated Ax--Kochen theorem establi
 shes an asymptotic version of Artin's conjecture when $p$ is large relativ
 e to $d$. In recent joint work with Lea Beneish\, we establish Artin's con
 jecture in degree 5 when $p > 5$ and in degree 7 when $p > 679$. In this t
 alk\, we will explore the ideas and techniques spanning nearly a century b
 ehind these results\, from the Lang--Weil theorem to effective Bertini the
 orems and parallel computing.\n
LOCATION:https://researchseminars.org/talk/UEAPS/59/
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