A prolongation of the question
[composition-of-polynomial-functions-which-gives-the-identity](https://mathoverflow.net/questions/83745/composition-of-polynomial-functions-which-gives-the-identity):
Let $f\_1,\ldots,f\_n, g\_1,\ldots, g\_n$ be polynomials in $\mathbb{Q}[X\_1,\ldots,X\_n]$ such that if $g=(g\_1,\ldots,g\_n)$ then $f\_i(g(x\_1,\ldots,x\_n))=x\_i$ for all $i=1,\ldots,n$. Does it follow that SOME $f\_i$ or $g\_j$ has degree 1?



polynomial maps
Of course not. For example, consider the automorphism of $\mathbb Q[x, y]$ given by $(x, y) \mapsto (x+y^2, y + (x+y^2)^2)$.


