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

摘要

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, Cl1+ and Cl1++, 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 Cl1+ 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 Cl1++ 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 Cl1+ which are contained in Cl-1 are explicitly described (Theorem 3).