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Artin's Conjecture for Forms of Degree 7 and 11
Author(s) -
Knapp Michael P.
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1017/s0024610700001721
Subject(s) - mathematics , conjecture , counterexample , diophantine equation , degree (music) , combinatorics , finite field , class (philosophy) , homogeneous , field (mathematics) , discrete mathematics , pure mathematics , physics , artificial intelligence , computer science , acoustics
A fundamental aspect of the study of Diophantine equations is that of determining when an equation has a local solution. Artin once conjectured (see the preface to [ 1 ]) that if k is a complete, discretely valued field with finite residue class field, then every homogeneous form of degree d in greater than d 2 variables whose coefficients are integers of k has a nontrivial zero. In this paper, we consider the case of this conjecture in which k is a p ‐adic field. Although a counterexample due to Terjanian [ 16 ] proved Artin's conjecture false in this situation, Ax and Kochen [ 2 ] have shown when [ k : Q p ] = n is finite, that given d , there exists a number p ( d , n ) such that Artin's conjecture is true provided that p is larger than p ( d , n ). Unfortunately, the methods of Ax and Kochen do not lead to explicit estimates for p ( d , n ). Cohen [ 5 ] found a method which determines the possible cardinalities of the residue class fields of all p ‐adic fields for which Artin's conjecture is false, and Brown [ 3 ] has used this to bound p ( d , 1), but this bound is so large that one feels that it must be possible to do better. Hence, it is still an interesting problem to obtain estimates on the size of p ( d , n ).