Some combinatorial questions about polynomial mappings (Q1358937)

Coordinate polynomials are elements of \(P_n= K[x_1, \dots, x_n]\), \(K\) a commutative field, which can be included in a generating set of \(P_n\) of cardinality \(n\). The authors prove that an endomorphism of \(P_2\) which fixes the set of coordinate polynomials is actually an automorphism. Assuming the Jacobian conjecture this result holds for arbitrary \(n\), too. Other results concern so-called test polynomials, for which \(\varphi (p)=p\), \(\varphi\in \text{End} P_n\), implies \(\varphi\) is an automorphism.