On the Complextity of the Differential-Algebraic Description of Analytic Complexity Classes

The objective of this paper is to trace the increase in the complexity of the description of classes of analytic complexity (introduced by the author in previous works) under the passage from the class $Cl_1$ to the class $Cl_2$. To this end, two subclasses, $Cl_1^+$ and $Cl_1^{++}$, of $Cl_2$ that are not contained in $Cl_1$ are described from the point of view of the complexity of the differential equations determining these subclasses. It turns out that $Cl_1^+$ has fairly simple defining relations, namely, two differential polynomials of differential order $5$ and algebraic degree $6$ (Theorem 1), while a criterion for a function to belong to $Cl_1^{++}$ obtained in the paper consists of one relation of order $6$ and five relations of order $7$, which have degree $435$ (Theorem 2). The “complexity drop” phenomenon is discussed; in particular, those functions in the class $Cl_1^+$ which are contained in $Cl_1$ are explicitly described (Theorem 3).