A prolongation of the question compositionofpolynomialfunctionswhichgivestheidentity: 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?
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Of course not. For example, consider the automorphism of $\mathbb Q[x, y]$ given by $(x, y) \mapsto (x+y^2, y + (x+y^2)^2)$. 


Here is another way to say what Angelo said. Pick any set of polynomials $\left(g_1,\ldots,g_n\right)$ such that it has Groebner basis $\left[x_1,\ldots,x_n\right]$ under lexicographical order $\left[x_n,\ldots,x_1\right]$. Then you can get the $f_j$'s from the cofactors (pdf). [This is merely a more constructive way of phrasing the answer, so that you may construct more examples for yourself easily, which is not as easy to do given Angelo's answer.] 

