论文标题
在代数关闭中排名
On rank in algebraic closure
论文作者
论文摘要
令$ {\ mathbf k} $为一个字段，$ q \ in {\ mathbf k} [x_1，\ ldots，x_s] $ $ $ d> 1的形式（均质多项式）的形式1. $ {\ mathbf k}那$ q = \ sum_ {i = 1}^r r_is_i $带有$ r_i，s_i \ in {\ mathbf k} [x_1，\ ldots，x_s] $ of度$ <d $。当$ {\ mathbf k} $在代数上关闭时，该排名本质上等同于$ {\ Mathbf k}^s $ in $ q的单数基因座的$ {\ mathbf k}^s $等同于$ q的定义，$ q，$ q，也称为$q。$q。$ q。 rk _ {\ Mathbf k}（q）$在$ rk _ {\ bar {\ mathbf k}}}}（q）$中，其中$ \ bar {\ mathbf k} $是$ {\ Mathbf k}的代数关闭。在这项工作之前，没有这样的约束（即使有效）以$ d> 4 $而闻名。此结果对数量点（$ {\ Mathbf k} $是一个数字字段）或素数（当$ {\ Mathbf k} = \ Mathbb Q $）的数量$ \ {q = 0 \} $假设$ rk _ {Q = 0 \} $ rk _ {{{{{{{\ mathbf k}（q）（q）（q）（当$ {\ Mathbf k} = \ altbb q $时（当$ {\ Mathbf k} = \ Mathbb Q $）对数字产生直接的后果。
Let $ {\mathbf k} $ be a field and $Q\in {\mathbf k}[x_1, \ldots, x_s]$ a form (homogeneous polynomial) of degree $d>1.$ The ${\mathbf k}$-Schmidt rank $rk_{\mathbf k}(Q)$ of $Q$ is the minimal $r$ such that $Q= \sum_{i=1}^r R_iS_i$ with $R_i, S_i \in {\mathbf k}[x_1, \ldots, x_s]$ forms of degree $<d$. When $ {\mathbf k} $ is algebraically closed, this rank is essentially equivalent to the codimension in $ {\mathbf k}^s $ of the singular locus of the variety defined by $ Q, $ known also as the Birch rank of $ Q. $ When $ {\mathbf k} $ is a number field, a finite field or a function field, we give polynomial bounds for $ rk_{\mathbf k}(Q) $ in terms of $ rk_{\bar {\mathbf k}} (Q) $ where $ \bar {\mathbf k} $ is the algebraic closure of $ {\mathbf k}. $ Prior to this work no such bound (even ineffective) was known for $d>4$. This result has immediate consequences for counting integer points (when $ {\mathbf k} $ is a number field) or prime points (when $ {\mathbf k} = \mathbb Q $) of the variety $ \{Q=0\} $ assuming $ rk_{\mathbf k} (Q) $ is large.
