Let $f_1,\ldots,f_{n-k} \in \mathbb{R}[x_1,\ldots,x_n]$ be polynomials of degree at most $d$ defining an algebraic set $A \subseteq \mathbb{C}^n$ which contains an irreducible component $V \subseteq A$ of dimension $k$. Is it possible to find polynomials $g_1,\ldots,g_r \in \mathbb{R}[x_1,\ldots,x_n]$ of degree at most $Cd$ (where the constant $C$ may depend on $n$ and $k$, but not on $d$) such that each $g_i$ vanishes on $V$ and such that the Jacobian matrix $(\partial_j g_i)_{i,j}$ of these polynomials has rank $n-k$ on at least one point of $V$?