On the Complexity of the Differential-Algebraic Description of Analytic Complexity Classesстатья
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Аннотация: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, Cl+
1 and Cl++
1 , 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 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 Cl1 are explicitly
described (Theorem 3).
Keywords: superposition of analytic functions, analytic complexity, differential polynomials