On Artin's Conjecture, I: Systems of Diagonal Forms

Abstract

As a special case of a well‐known conjecture of Artin, it is expected that a system of R additive forms of degree k, say [formula] with integer coefficients aij, has a non‐trivial solution in Qp for all primes p whenever [formula] Here we adopt the convention that a solution of (1) is non‐trivial if not all the xi are 0. To date, this has been verified only when R=1, by Davenport and Lewis [4], and for odd k when R=2, by Davenport and Lewis [7]. For larger values of R, and in particular when k is even, more severe conditions on N are required to assure the existence of p‐adic solutions of (1) for all primes p. In another important contribution, Davenport and Lewis [6] showed that the conditions [formula] are sufficient. There have been a number of refinements of these results. Schmidt [13] obtained N≫R2k3 log k, and Low, Pitman and Wolff [10] improved the work of Davenport and Lewis by showing the weaker constraints [formula] to be sufficient for p‐adic solubility of (1). A noticeable feature of these results is that for even k, one always encounters a factor k3 log k, in spite of the expected k2 in (2). In this paper we show that one can reach the expected order of magnitude k2. 1991 Mathematics Subject Classification 11D72, 11D79.

Local solvability of diagonal equations (again)

“…Next we indicate how to obtain the same bound on Γ (R, m) for a general K as that given in [BG,Theorem 3] for K = Q p (when R ≥ 2 this bound is slightly better than that stated in Theorem B). More precisely, we explain how to modify the statements of the results used in the proof in [BG] so that they apply to the general situation, that is, to the situation where "systems" are systems of equations or congruences with coefficients in O and "solutions" are solutions with entries in O.…”

Diagonal equations of different degrees over p-adic fields

“…Recently, Brüdern & Godinho ( [2], [3]) have proven this case of Artin's conjecture for most even values of k. In particular, they have shown that Γ * p (k, k) ≤ 2k 2 + 1 except possibly when either k = p τ (p − 1) with p prime and τ ≥ 1, or k = 3 · 2 τ . Even in these situations, however, they have shown that Γ * p (k, k) ≤ 8k 2 .…”

“…Much is known about this problem in the situation where k = n. Davenport & Lewis [7] have shown that (2) Γ * (k, k) ≤ 2k 2 + 1, k odd, 7k 3 , k even.…”

Simultaneous diagonal equations over p-adic fields

“…From (4), (2), and (7) it now follows that Γ (R, k) ≤ Rk ·Φ(R, k, γ) ≤ Rk ·(Φ(R, k, e)+1) γ/e ≤ (Rk) 2γ/e+1 ≤ (Rk) 2τ +5 .…”

“…The Chevalley-Warning theorem (see [2,Lemma 4]) states that any system of homogeneous polynomials over a finite field has a non-trivial zero if the number of variables exceeds the sum of the polynomials' degrees. In the special case of systems of diagonal equations, the Chevalley-Warning theorem gives…”