Change of variables of a polynomial

Question

Can someone help me please to solve this problem: we consider $V(Y)$ a polynomial in $ R[Y_1,Y_2,..,Y_d] $ I want to prove that there existe an affine change of variables $ Y=AX+B,X=(X_1,X_2,..,X_d)$ and there exist $n=n(V)\in N $ such that : 1) for any $j∈${$1,..,n(V)$} we have $∂_{X_j}V=$ complex constant
2)for any $j∈${$n(V)+1,..,d$} there exist $f_j∈R[X_{j+1},..,X_d]$ such that $X_j+f_j(X_{j+1},..,X_d)∈ Span ${$∂_XV,α∈N^d,|α|≥1$} indication: consider the linear map $∂_{Y_j},j=1,..,d$ which send the finite vector space $F_p=Span${$∂_XV,α∈N^d,|α|≥p$} to $F_{p+1}$ and construct the coordonnates $X_j $by reverse induction in a Jordan approch.