On Buzzard's Theoren

Abstract

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