Geodesic
On the Complextity of the Differential-Algebraic Description of Analytic Complexity Classes
Matematičeskie zametki, Tome 105 (2019) no. 3, pp. 323-331.

Voir la notice de l'article provenant de la source Math-Net.Ru

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).
Keywords: superposition of analytic functions, analytic complexity, differential polynomials. 
@article{MZM_2019_105_3_a0,
     author = {V. K. Beloshapka},
     title = {On the {Complextity} of the {Differential-Algebraic} {Description} of {Analytic} {Complexity} {Classes}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {323--331},
     publisher = {mathdoc},
     volume = {105},
     number = {3},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2019_105_3_a0/}
}
TY  - JOUR
AU  - V. K. Beloshapka
TI  - On the Complextity of the Differential-Algebraic Description of Analytic Complexity Classes
JO  - Matematičeskie zametki
PY  - 2019
SP  - 323
EP  - 331
VL  - 105
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2019_105_3_a0/
LA  - ru
ID  - MZM_2019_105_3_a0
ER  - 
%0 Journal Article
%A V. K. Beloshapka
%T On the Complextity of the Differential-Algebraic Description of Analytic Complexity Classes
%J Matematičeskie zametki
%D 2019
%P 323-331
%V 105
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2019_105_3_a0/
%G ru
%F MZM_2019_105_3_a0
V. K. Beloshapka. On the Complextity of the Differential-Algebraic Description of Analytic Complexity Classes. Matematičeskie zametki, Tome 105 (2019) no. 3, pp. 323-331. http://geodesic.mathdoc.fr/item/MZM_2019_105_3_a0/

[1] A. Ostrovski, “Über Dirichletsche Raihen und algebraishe Differentialgleichungen”, Math. Z., 8:3-4 (1920), 241–298 | DOI | MR

[2] A. G. Vitushkin, “13-ya problema Gilberta i smezhnye voprosy”, UMN, 59:1 (355) (2004), 11–24 | DOI | MR | Zbl

[3] V. K. Beloshapka, “Analytic Complexity of Functions of Two Variables”, Russ. J. Math. Phys., 14:3 (2007), 243–249 | DOI | MR | Zbl

[4] V. A. Krasikov, T. M. Sadykov, “Ob analiticheskoi slozhnosti diskriminantov”, Analiticheskie i geometricheskie voprosy kompleksnogo analiza, Sbornik statei, Tr. MIAN, 279, MAIK «Nauka/Interperiodika», M., 2012, 86–101 | MR

[5] V. K. Beloshapka, “Three families of functions of complexity one”, J. Sib. Fed. Univ. Math. Phys., 9:4 (2016), 416–426 | DOI

[6] E. A. Kalinina, A. Yu. Uteshev, Teoriya isklyucheniya, NIIKh SPbGU, 2002