A parachute for the degree of a polynomial in algebraically independent ones

We give a simpler proof as well as a generalization of the main result of an article of Shestakov and Umirbaev. This latter article being the first of two that solve a long-standing conjecture about the non-tameness, or 'wildness', of Nagata's automorphism. As corollaries we get interesting informations about the leading terms of polynomials forming an automorphism in any dimension and reprove the tameness of automorphisms in dimension two.