Artin’s Conjecture for Forms of Degree 7 and 11

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 # 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 padic 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). Previous to Ax and Kochen’s proof, several results of this kind were already known. Hasse [9] showed that p(2, n) ̄ 1 for all n, and Demyanov [6] (when the characteristic of the residue field is not 3) and Lewis [13] proved that p(3, n) ̄ 1. That is, Artin’s conjecture is true for d ̄ 2 and d ̄ 3. Furthermore, Birch and Lewis [4] and Laxton and Lewis [11] showed the existence of p(5, n), p(7, n) and p(11, n), but were unable to estimate their values. More recently, Leep and Yeomans [12] obtained the bound p(5, n)% 43. In this note, we will show how a theorem due to Schmidt can be combined with the method of Laxton and Lewis to obtain upper bounds for p(7, n) and p(11, n). In particular, in Section 3 we prove the following theorem.