Two improvements in Brauer's theorem on forms

2024/01/04 by Arthur Bik, Bik, Arthur, Jan Draisma +3 · 2 citations
Computer Science · Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Meromorphic and Entire Functions #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2401.02067

openalex publication_date 2024/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let k be a Brauer field, that is, a field over which every diagonal form in sufficiently many variables has a nonzero solution; for instance, k could be an imaginary quadratic number field. Brauer proved that if f₁, …, f_r are homogeneous polynomials on a k-vector space V of degrees d₁, …, d_r, then the variety Z defined by the f_i's has a non-trivial k-point, provided that dimV is sufficiently large compared to the d_i's and k. We offer two improvements to this theorem, assuming k is infinite. First, we show that the Zariski closure of the set Z(k) of k-points has codimension

Cited by

• Polynomial Bounds for Birch's Theorem
• Density of solutions for systems of forms

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