elementary superfunctions

[bo198214]

Henryk, it seems to me that such a case can be obtained from the example 5 of the Table 1 of our article

Yes that is right, \( f \) is polynomially conjugated to \( x^2 \), i.e. there exists a polynomial \( P \) such that \( f(P(x))=P(x^2) \), and as we know a superfunction of \( x^2 \) is \( e^{2^x} \), we know that a superfunction of \( f \) is \( P(e^{2^x}) \).

But we dont have a decision criteria when a given polynomial \( f \) is conjugated to some \( x^n \).

Particularly all polynomials without real fixed point, e.g. \( x^2+1 \), seem not be (real) conjugated to some \( x^n \).