A Note on Artin's Diophantine Conjecture
Canadian mathematical bulletin, Tome 13 (1970) no. 1, pp. 119-120
Voir la notice de l'article provenant de la source Cambridge University Press
A well known theorem of Hasse [1] says that every quadratic form in at least 5 variables over the field Q p of p-adic numbers has a nontrivial zero. This fact has led Artin to make the conjecture(C): "Every form over Q p of degree d in n > d 2 variables has a non-trivial zero." However, a counterexample has been provided by Terjanian [2] in the case d=4.The case d=2 is distinguished by the fact that every quadratic form may be "diagonalized", i.e., assumed to be of the type Σ a i X 2 i . One is therefore led to the weaker conjecture(C): "Every form f=Σ a i X d i over Q p in n > d 2 variables has a nontrivial zero in Q p,"which still generalizes Hasse's theorem.
Maxwell, George. A Note on Artin's Diophantine Conjecture. Canadian mathematical bulletin, Tome 13 (1970) no. 1, pp. 119-120. doi: 10.4153/CMB-1970-024-3
@article{10_4153_CMB_1970_024_3,
author = {Maxwell, George},
title = {A {Note} on {Artin's} {Diophantine} {Conjecture}},
journal = {Canadian mathematical bulletin},
pages = {119--120},
year = {1970},
volume = {13},
number = {1},
doi = {10.4153/CMB-1970-024-3},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1970-024-3/}
}
[1] 1. Hasse, H., Darstellbarkeit von Zahlen durch Quadratische Formen, J. f. reine u. angew. Math. 153 (1923), 113-130. Google Scholar
[2] 2. Terjanian, G., Un contre-exemple à une conjecture d' Artin, Comptes Rendus de l' Acad. Sci. Paris 262 (1966), A612. Google Scholar
[3] 3. Chevalley, C., Démonstration d'une hypothèse de M. Artin, Abh. Math. Sem. Hambur. 11 (1935), 73-75. Google Scholar
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