On rank in algebraic closure

Abstract

Let \( {{\textbf{k}}}\) be a field and \(Q\in {{\textbf{k}}}[x_1, \ldots , x_s]\) a form (homogeneous polynomial) of degree \(d>1.\) The \({{\textbf{k}}}\)-Schmidt rank \(\text {rk}_{{\textbf{k}}}(Q)\) of Q is the minimal r such that \(Q= \sum _{i=1}^r R_iS_i\) with \(R_i, S_i \in {{\textbf{k}}}[x_1, \ldots , x_s]\) forms of degree \(<d\). When \( {{\textbf{k}}}\) is algebraically closed and \( \text {char}({{\textbf{k}}})\) doesn’t divide d, this rank is closely related to \( \text {codim}_{{\mathbb {A}}^s} (\nabla Q(x) = 0)\) - also known as the Birch rank of Q. When \( {{\textbf{k}}}\) is a number field, a finite field or a function field, we give polynomial bounds for \( \text {rk}_{{\textbf{k}}}(Q) \) in terms of \( \text {rk}_{{\bar{{{\textbf{k}}}}}} (Q) \) where \( {\bar{{{\textbf{k}}}}} \) is the algebraic closure of \( {{\textbf{k}}}. \) Prior to this work no such bound (even ineffective) was known for \(d>4\). This result has immediate consequences for counting integer points (when \( {{\textbf{k}}}\) is a number field) or prime points (when \( {{\textbf{k}}}= {\mathbb {Q}}\)) of the variety \( (Q=0) \) assuming \( \text {rk}_{{{\textbf{k}}}} (Q) \) is large.