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

Abstract

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 Cl₁ to the class Cl₂. To this end, two subclasses, Cl₁⁺ and Cl₁⁺⁺, of Cl₂ that are not contained in Cl₁ 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 Cl₁ are explicitly described (Theorem 3).