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SUMMARY:Chris Keyes (King's College London)
DTSTART:20250305T160000Z
DTEND:20250305T170000Z
DTSTAMP:20260418T063835Z
UID:LNTS/152
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/LNTS/152/">T
 owards Artin's conjecture on $p$-adic quintic forms</a>\nby Chris Keyes (K
 ing's College London) as part of London number theory seminar\n\nLecture h
 eld in Room 505\, Department of Mathematics (25 Gordon St)\, University Co
 llege London.\n\nAbstract\nLet $K/\\mathbb{Q}_p$ be a finite extension wit
 h 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 d
 ue 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 l
 arge 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 prov
 e 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.\
 n
LOCATION:https://researchseminars.org/talk/LNTS/152/
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