Test polynomials (Q1973360)

A polynomial \(p\in\mathbb C[x_1,\dots ,x_n]\) is called a test polynomial (resp. a strong test polynomial) if for any automorphism \(\psi\) and for any endomorphism \(\phi\) of \(\mathbb C[x_1,\dots ,x_n]\) the equality \(\phi(p) =\psi(p)\) implies that \(\phi\) is an automorphism (resp. \(\phi=\psi\)). The main results of the paper are: (1) The generic polynomial \(p\) of degree \(>n\) is a strong test polynomial in the class of all monomorphisms of \(\mathbb C[x_1,\dots ,x_n]\), (2) For every \(D>0\) there is a strong test polynomial \(p\) of degree \(>D\), (3) For \(n=2\) a generic polynomial of degree \(>3\) is a strong test polynomial and a generic polynomial of degree \(\leq 3\) is not a test polynomial.