On Buzzard's Theoren

Over any infinite field we prove existence of polynomial automorphism with prescribed differentials at points. More precisely, let $\text{K}$ be an infinite field and $a_1,\dots ,a_k;$ $b_1,\dots ,b_k$ be two sequences of different points in~${\text{K}}^n$, $n\ge 2$. For any sequence $L_1,\dots ,L_k\in SL(n,\text{K})$ there exists a polynomial automorphism $\mathrm{\Phi }:{\text{K}}^n\to {\text{K}}^n$ with jacobian one such that $\mathrm{\Phi }(a_i)=b_i$ and $d_{a_i}\mathrm{\Phi }=L_i$ for $i=1,\dots ,k\mathrm{.}$