Anisotropic forms modulo p²

Artin's conjecture for septic and unidecic forms

“…It transpires that the methods employed in our proof of Theorem 1 may be applied to address the solubility of congruences modulo p 2 , for prime numbers p, thereby improving a theorem of Chakri and Hanine (see Theorem 3.1 of [11]) that makes explicit an earlier conclusion of Ax and Kochen [3]. In §4 we prove the following theorem.…”

“…, x n ] be homogeneous of degree d, and suppose that n ≥ 2d and p > 1 2 (3d 4 − 4d 3 + 5d 2 ). An inspection of the proof of [11,Theorem 3.1] reveals that the conclusion of Theorem 2 follows at once whenever f * fails to be absolutely irreducible, even in the absence of the hypothesis on p. Henceforth, therefore, we may suppose that f * is absolutely irreducible. Given our hypothesis on p, we may apply Lemma 5(i) to f * with δ = d to deduce that a slice ξ ∈ F 3n+1 p exists for which f * | ξ is absolutely irreducible.…”

“…The proof of Theorem 2. We now turn our attention to the refinement of Theorem 3.1 of Chakri and Hanine [11] embodied in Theorem 2 above. Let f ∈ Z p [x 0 , .…”