On rank in algebraic closure

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.