Michael P. Knapp

Systems of Diagonal Equations Over p-Adic Fields

Let [ ] be a [pfr ]-adic field, and consider the system F = ( F 1 ,…, F R ) of diagonal equations [formula here] with coefficients in [ ]. It is an interesting problem in number theory to determine when such a system possesses a nontrivial [ ]-rational solution. In particular, we define Γ*( k , R , [ ]) to be the smallest natural number such that any system of R equations of degree k in N variables with coefficients in [ ] has a nontrivial [ ]-rational solution provided only that N [ges ]Γ*( k , R , [ ]). For example, when k = 1, ordinary linear algebra tells us that Γ*(1, R , [ ]) = R + 1 for any field [ ]. We also define Γ*( k , R ) to be the smallest integer N such that Γ*( k , R , ℚ p ) [les ] N for all primes p .

On systems of diagonal forms

In this paper we consider systems of diagonal forms with integer coefficients in which each form has a different degree. Every such system has a nontrivial zero in every p-adic field [Qp] provided that the number of variables is sufficiently large in terms of the degrees. While the number of variables required grows at least exponentially as the degrees and number of forms increase, it is known that if p is sufficiently large then only a small polynomial bound is required to ensure zeros in [Qp]. In this paper we explore the question of how small we can make the prime p and still have a polynomial bound. In particular, we show that we may allow p to be smaller than the largest of the degrees.

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.

On systems of diagonal forms II

Given a system of diagonal forms over ℚ p , we ask how many variables are required to guarantee that the system has a nontrivial zero. We show that if the prime p satisfies p > (largest degree) − (smallest degree) + 1, then there is a bound on the sufficient number of variables which is a polynomial in the degrees of the forms.

On Weakly Complete Sequences Formed by the Greedy Algorithm

Abstract. In this article, we consider increasing sequences of positive integers defined in the following manner. Let the initial terms a1 and a2 be given, and for any define to be the smallest integer greater than which cannot be written as a sum of (distinct) previous terms of the sequence. For various parametrized choices of the initial terms, we determine precisely the terms of the sequences obtained by this method. We also conjecture that for all choices of the initial terms, even in a more general setting, the terms of sequences defined in this manner have interesting patterns.