Zeros of p-adic forms

Abstract

We show that all p-adic quintic forms in at least n > 4562911 variables have a non-trivial zero. We also derive a new result concerning systems of cubic and quadratic forms.

Improvements in Birch’s theorem on forms in many variables

“…In particular, when Theorem 1.2 is applied to non-singular quintic forms, it suffices to check the solubility over Q p for primes p 13. In this range, the best result we have is due to Zahid [16], who establishes that ν 5 (p) 4562912 for p 13.…”

Quartic Forms in Many Variables

“…Firstly, we can extract better estimates from Wooley's proof for specifc d. Secondly, Heath-Brown [7] considerably improved these for a single quartic by establishing v(4) ≤ 4220. His proof has been adapted by Zahid [11] to show v(5) ≤ 4562911 (Note that in this case the conjecture has been confirmed if p > 7. See [5]).…”

“…Several results address specific systems of forms. To put the next result into perspective it suffices to know that v(3, 3) ≤ 213 can be derived by combing [7] and [11].…”

Rational lines on cubic hypersurfaces

“…We now choose an unramified quadratic extension L of Q p . As p ě 19, the residue class field of L over Q p has cardinality p 2 ą 293, and the existence of a non-trivial solution y of (3.1) over L then follows from work of Zahid [28,Theorem 1.1]. Thus again, x 0 and y span a line over L, which can be pulled back to Q p by [26, Lemma 2.2].…”