Differentially real, homogeneous, and additive functions (Q1336229)

A function \(u(z)\) is said to be differentially algebraic (DA) if it satisfies a non-trivial algebraic differential equation (ADE), that is, an equation of the form \(P(z,y(z),\dots, y^{(n)}(z))= 0\), where \(P\) is a polynomial with complex coefficients in its \(n+2\) variables. It is called respectively differentially real, differentially homogeneous, and differentially additive, if the respective function \(\bar u(\bar z)\), \(c(z)\), \(u(z)+ c\) (\(c\) is any complex constant) satisfies every ADE which is satisfied by \(u(z)\). The author proves the following statements: If \(f\) is DA and real on \(\mathbb{R}\) and \(g\) is any differentially real entire function, then the composition \(f\circ g\) is differentially real. If \(u\) is a non-constant DA entire function with no zeros on the complex plane, then \(u\) is differentially homogeneous. Suppose \(g\) is a DA entire function algebraically dependent over the differential field generated by \(g'\) and suppose further that the order of any minimal polynomial for \(g\) is \(n\). Then any minimal polynomial for \(e^ g\) is of order \(n+1\).