Title: Two improvements in Brauer's theorem on forms Show Chinese title

Title: 关于型的布劳尔定理的两项改进

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. Show Chinese abstract

Abstract: 设 $k$ 为一个 Brauer 域，即在一个域上，足够多个变量的对角形式有非零解；例如，$k$ 可以是虚二次数域。 布劳尔证明了，如果 $f_1, \ldots, f_r$ 是定义在 $k$-向量空间 $V$ 上的齐次多项式，且它们的次数分别为 $d_1, \ldots, d_r$，那么当 $\dim{V}$ 充分大（相对于 $d_i$ 和 $k$ 的大小），由这些 $f_i$ 定义的簇 $Z$ 存在一个非平凡的 $k$-点。 我们提出对该定理的两项改进，假设$k$是无限的。 首先，我们证明集合$Z(k)$的Zariski闭包中$k$-点集的余维数为$<C$，其中$C$是仅依赖于$d_i$和$k$的常数。 其次，我们证明了如果在 $d_i$ 和 $k$ 的意义下 $f_i$ 的强度足够大，则 $Z(k)$ 实际上在 $Z$ 中是 Zariski 致密的。 这些证明依赖于 Ananyan 和 Hochster 关于高强度多项式最近的工作。