On \(p\)-adic zeros of forms
type: Item
instance of: scholarly article
title: On \(p\)-adic zeros of forms
zbMATH Open document ID: 0531.10027
DOI: 10.1016/0022-314X(84)90066-0
author: W. Dale Brownawell
published in: Journal of Number Theory
publication date: 1984-01-01T00:00:00Z
review text: The author sharpens the remarkable result of \textit{G. I. Arkhipov} and \textit{A. A. Karatsuba} [Math. USSR, Izv. 19, 231--240 (1982); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 45, 948--961 (1981; Zbl 0476.10018)] that the number of variables possible in a form having only the trivial p-adic zero has an exponential lower bound with respect to the degree (and so is not even limited to polynomial growth). His estimate is stronger than Arkhipov and Karatsuba's recently announced improvement [Usp. Mat. Nauk 37, No. 5(227), 161--162 (1982; Zbl 0508.10012)] of their original results; their improvement is in turn stronger than the result of \textit{D. J. Lewis} and \textit{H. L. Montgomery} [Mich. Math. J. 30, 83--87 (1983; Zbl 0531.10026)].
Mathematics Subject Classification ID: 11E95
Mathematics Subject Classification ID: 11E76
Mathematics Subject Classification ID: 11D72
zbMATH DE Number: 3841966
zbMATH Keywords: higher degree forms
zbMATH Keywords: Artin conjecture
zbMATH Keywords: \(p\)-adic zero
zbMATH Keywords: exponential lower bound
reviewed by: M. M. Dodson
MaRDI profile type: Publication
cites work: ON LOCAL REPRESENTATION OF ZERO BY A FORM
cites work: Q5514706
cites work: Q5581789
cites work: Q5577214
cites work: On zeros of \(p\)-adic forms
cites work: An extension of Chevalley's theorem to congruences modulo prime powers
cites work: Q5343475
cites work: Formes p-adiques anisotropes.
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description: scientific article; zbMATH DE number 3841966