On rank in algebraic closure

Let k be a field and Q∈ k[x₁, … , x_s] a form (homogeneous polynomial) of degree d> 1. The k -Schmidt rank rk _k(Q) of Q is the minimal r such that Q=∑i=1rRiSi with R_i, S_i∈ k[x₁, … , x_s] forms of degree < d . When k is algebraically closed and char (k) doesn’t divide d, this rank is closely related to codimAs(∇Q(x)=0) - also known as the Birch rank of Q. When k is a number field, a finite field or a function field, we give polynomial bounds for rk _k(Q) in terms of rk _k¯(Q) where k¯ is the algebraic closure of 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 k is a number field) or prime points (when k= Q) of the variety (Q= 0) assuming rk _k(Q) is large.