On functions of finite analytical complexity

We construct examples of polynomials and analytic functions of any predetermined finite analytical complexity $n$. We obtain an estimate of the order of derivative of the differential-algebraic criteria for membership in the class $Cl_n$ of functions of analytical complexity not higher than $n$. We find uniform estimates for finite values $d_n$ of the analytic spectrum $\{d_n\}$ for systems of differential-algebraic equations of fixed order of derivative $\delta$.