{"id":3395,"date":"2024-02-21T16:53:17","date_gmt":"2024-02-22T00:53:17","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=3395"},"modified":"2024-02-21T16:53:17","modified_gmt":"2024-02-22T00:53:17","slug":"a-group-theoretic-ax-katz-theorem-pete-l-clark-university-of-georgia","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/a-group-theoretic-ax-katz-theorem-pete-l-clark-university-of-georgia\/","title":{"rendered":"A Group-Theoretic Ax-Katz Theorem (Pete L. Clark, University of Georgia)"},"content":{"rendered":"

Title:<\/strong> A Group-Theoretic Ax-Katz Theorem<\/p>\n

Speaker: <\/strong>Pete L. Clark, University of Georgia<\/p>\n

Abstract: <\/strong>The Chevalley-Warning Theorem is a result from 1935 asserting that the number of solutions to a low degree polynomial system over a finite field is divisible by the characteristic of the field.\u00a0 It is an important result — it includes a conjecture of Artin and Dickson from the 1920’s — but it is not difficult to prove: the original proof is about three pages.\u00a0 In 1964 James Ax gave a completely elementary ten line proof<\/b>.\u00a0\u00a0 In the same paper, Ax showed that as the number and degrees of the polynomials are held fixed and the number of variables increases, not only is the size of the solution set divisible by p but by higher and higher powers of p.\u00a0 The best possible p-adic divisibility here was given in 1971 by Nicholas Katz.\u00a0 Katz’s proof is at a much higher level: you need specialist knowledge in the right subfields of number theory to understand it.\u00a0 Simpler proofs were found later, but none fulfills the fantasy of generalizing Ax’s ten line proof of Chevalley-Warning.<\/p>\n

A 2021 work of Aichinger-Moosbauer develops a fully fledged calculus of finite differences for maps between commutative groups and uses it to give a purely group-theoretic generalization of Chevalley-Warning. Nicholas Triantafillou and I have used and extended this work: up to a few black boxes (where most of the content is indeed hidden) we give a ten line proof of a group-theoretic analogue of Ax-Katz that “qualitatively fulfills my fantasy.”<\/div>\n
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In (North)west Philadelphia was Pete L. Clark born and raised.\u00a0 He received undergraduate and masters degrees from the University of Chicago and a PhD from Harvard University.\u00a0 He has worked in the Mathematics Department at the University of Georgia since 2006, where he was the Graduate Coordinator from 2016-2019 and where he is now the Principal Honors Advisor.\u00a0 When time permits he is an avid reader, and his favorite authors include Ralph Ellison, Jonathan Franzen, Kazuo Ishiguro, Carmen Maria Machado and Lorrie Moore.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Title: A Group-Theoretic Ax-Katz Theorem Speaker: Pete L. 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