Zeros of p-adic forms

2008/05/31 by D. R. Heath-Brown · 1 citation
Mathematics · #Advanced Mathematical Identities #Algebra over a field #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Quartic function #Quintic function #Type (biology) #Variable (mathematics) #math.NT #msc:11D72 #msc:11D88

paper · pdf · doi:10.1112/plms/pdp043

Misprints and other errors corrected, leading to a small improvement

arxiv created 2009/09/14 · openalex publication_date 2009/11/12 · arxiv updated 2014/02/26 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

A variant of Brauer's induction method is developed. It is shown that quartic p-adic forms with at least 9127 variables have non-trivial zeros, for every p. For odd p considerably fewer variables are needed. There are also subsidiary new results concerning quintic forms, and systems of forms.

Cited by

• Ax–Kochen theorem [wikipedia]

Related

• Quartic Forms in Many Variables
• p-adic Zeros of Quintic Forms
• Artin's Conjecture on Zeros of p-Adic Forms
• Zeros of Systems of \mathfrak p-adic Quadratic Forms
• The Number of Zeros of Quadratic Forms with Two Terms