Pairs of additive forms of odd degrees

Exact values of the function Γ⁎(k)

“…For the most part, the proof of this theorem will proceed along the same lines as the proof of Lemma 1 of [9]. For specific k and p, the problem reduces to finding a nonsingular solution of a particular congruence equation.…”

“…We note here that Bovey's method was not used while proving the lemma in [9], and represents a significant computational improvement. This is because checking a particular pair of k and p via Bovey's method is much faster than checking the same pair by testing every possible congruence.…”

“…While studying an aspect of Artin's conjecture relating to systems of equations [9], the author was led to investigate the values of Γ * (k) for odd values of k. This work involved evaluating Γ * (k) for small odd k, with the following results.…”

On Artin’s conjecture: Linear slices of diagonal hypersurfaces

“…When k 1 = k 2 is even, Brüdern and Godinho [5] confirmed the conjecture for many k (see also Kränzlein [17]). Knapp [16] then showed that the conjecture also holds when R = 2 and k 1 = k 2 are both odd, while Wooley [27] showed that when R = 2, k 1 = 2 and k 2 = 3 then S ≥ 11 suffices to ensure p-adic non-trivial solutions. For larger R little is known (see [4,10,20]).…”

Artin's conjecture and systems of diagonal equations

“…There has been much work in the recent literature concerning systems of diagonal equations and their relation to AC (see [6,7,11,12,13,14,15]). In particular, it is known that AC holds for pairs of diagonal forms of equal degree k, except possibly when k takes the form p τ (p − 1) or 3 · 2 τ (p = 2), and that AC holds also for pairs of diagonal forms of distinct odd degrees.…”