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August 2020 Infinitely Many Counterexamples to a Conjecture of Norton
Hemar Godinho, Michael P. Knapp
Michigan Math. J. 69(3): 533-543 (August 2020). DOI: 10.1307/mmj/1596700817

Abstract

For any positive integer k, we define Γ(k) to be the smallest number s such that every diagonal form a1x1k+a2x2k++asxsk in s variables with integer coefficients must have a nontrivial zero in every p-adic field Qp. An old conjecture of Norton is that we should have Γ(k)1(modk) for all k. For many years, Γ(8)=39 was the only known counterexample to this conjecture, and in recent years two more counterexamples have been found. In this article, we produce infinitely many counterexamples to Norton’s conjecture.

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Hemar Godinho. Michael P. Knapp. "Infinitely Many Counterexamples to a Conjecture of Norton." Michigan Math. J. 69 (3) 533 - 543, August 2020. https://doi.org/10.1307/mmj/1596700817

Information

Received: 20 June 2018; Revised: 5 December 2018; Published: August 2020
First available in Project Euclid: 6 August 2020

MathSciNet: MR4132602
Digital Object Identifier: 10.1307/mmj/1596700817

Subjects:
Primary: 11D72
Secondary: 11D88

Rights: Copyright © 2020 The University of Michigan

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Vol.69 • No. 3 • August 2020
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