Abstract
It is proved that, for every rational function of two variables P(x, y) of analytic complexity one, there is either a representation of the form f(a(x) + b(y)) or a representation of the form f(a(x)b(y)), where f(x), a(x), b(x) are nonconstant rational functions of a single variable. Here, if P(x, y) is a polynomial, then f(x), a(x), and b(x) are nonconstant polynomials of a single variable.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Ostrowski, “Über Dirichletsche Reihen und algebraische Differentialgleichungen,” Math. Z. 8, 241–298, (1920).
D. Yu. Grigoriev, “Additive Complexity in Directed Computations,” Theoret. Comput. Sci. 19 (1), 39–67 (1982).
L. Blum, S. Smale, and M. Shub, “On a Theory of Computation and Complexity over the Real Numbers: NP-Completeness, Recursive Functions and Universal Machines,” Bull. Amer. Math. Soc. (N.S.) 21 (1), 1–46 (1989).
A. G. Vitushkin, “Hilbert’s thirteenth problem and related questions,” Uspekhi Mat. Nauk 59 1 (355), 11–24 (2004) [Russian Math. Surveys 59 (1), 11–25 (2004)].
V. K. Beloshapka, “Analytic Complexity of Functions of Two Variables,” Russ. J. Math. Phys. 14 (3), 243–249 (2007).
V. K. Beloshapka, “Analytical Complexity: Development of the Topic,” Russ. J. Math. Phys. 19 (4), 428–439 (2012).
V. A. Krasikov and T. M. Sadykov, “On the analytic complexity of discriminants,” in: Tr. Mat. Inst. Steklova 279 (2012), Analiticheskie i Geometricheskie Voprosy Kompleksnogo Analiza, pp. 86–101 [Proc. Steklov Inst. Math. 279 (1), 78–92 (2012)].
V. K. Beloshapka, “A Seven-Dimensional Family of Simple Harmonic Functions,” Mat. Zametki 98 (6), 803–808 (2015) [Math. Notes 98 (6), 867–871 (2015)].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Stepanova, M. On rational functions of first-class complexity. Russ. J. Math. Phys. 23, 251–256 (2016). https://doi.org/10.1134/S1061920816020102
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1061920816020102