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$ p$-adic zeros of quintic forms


Author: Jan H. Dumke
Journal: Math. Comp. 86 (2017), 2469-2478
MSC (2010): Primary 11D88; Secondary 11D72, 11E76
DOI: https://doi.org/10.1090/mcom/3182
Published electronically: March 3, 2017
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Abstract: It is shown that a quintic form over a $ p$-adic field with at least $ 26$ variables has a non-trivial zero, providing that the cardinality of the residue class field exceeds $ 9$.


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Additional Information

Jan H. Dumke
Affiliation: Mathematisches Institut, Bunsenstrasse 3-5, 37073 Göttingen, Germany
Email: jdumke@uni-math.gwdg.de

DOI: https://doi.org/10.1090/mcom/3182
Keywords: Artin's conjecture, $p$-adic forms, forms in many variables
Received by editor(s): February 4, 2014
Received by editor(s) in revised form: March 27, 2014, August 14, 2014, January 24, 2016, and May 5, 2016
Published electronically: March 3, 2017
Article copyright: © Copyright 2017 American Mathematical Society