Number Theory Seminar - Christopher Keyes
| Dates: | 18 March 2025 |
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| Times: | 15:00 - 16:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Christopher Keyes (KCL)
Title: Towards Artin's conjecture on p-adic quintic forms
Abstract: Let K be a p-adic field whose residue field has q elements and suppose f is a homogeneous polynomial of degree d in n+1 variables over K. A conjecture, originally due to Artin, states that when d is prime and n is at least 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 whenever q is at least 7. Our methods include both a refinement to a geometric approach of Leep and Yeomans (who showed q at least 47 suffices) and a significant computational component.
Room: Frank Adams 1
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Frank Adams 1
Alan Turing Building
Manchester
