Two improvements in Brauer's theorem on forms

TL;DR

The paper advances Brauer’s theorem on systems of homogeneous forms over Brauer fields by proving two substantial refinements. It establishes a uniform codimension bound on the Zariski closure of the $k$-rational points of the solution variety and, under a high-strength hypothesis on the defining forms, shows that the $k$-points are Zariski dense in the variety. The approach fuses strength theory (Ananyan–Hochster) with a careful reduction to a normal form, using a two-track argument that treats characteristic 0 and characteristic $p$ separately; the core reduction to a key proposition about good subspaces drives the density results and the corollary for diagonal equations. The work yields concrete solvability criteria for diagonal forms in many variables and connects Brauer-field phenomena to universality questions for high-strength tensors, with implications for still-unsettled questions about field extensions and the Stillman/strength program. Overall, the results quantify abundance of $k$-points and provide a robust framework for extending Brauer-type theorems to Brauer fields and beyond."

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 $f_1, \ldots, f_r$ are homogeneous polynomials on a $k$-vector space $V$ of degrees $d_1, \ldots, d_r$, then the variety $Z$ defined by the $f_i$'s has a non-trivial $k$-point, provided that $\dim{V}$ 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 $<C$, where $C$ is a constant depending only on the $d_i$'s and $k$. And second, we show that if the strength of the $f_i$'s is sufficiently large in terms of the $d_i$'s and $k$, then $Z(k)$ is actually Zariski dense in $Z$. The proofs rely on recent work of Ananyan and Hochster on high strength polynomials.

Theorems & Definitions (67)

• Definition 1.1
• Theorem 1.2: Brauer
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• proof
• Proposition 2.1: Peck, Peck
• Proposition 2.2
• proof
• Corollary 2.3