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Short communications

On local invertibility of functions of an $h$-complex variable

V. A. Pavlovsky, I. L. Vasiliev

Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

Abstract: The theory of functions of an $h$-complex variable is an alternative to the usual theory of functions of a complex variable, obtained by replacing the rules of multiplication. This change leads to the appearance of zero divisors on the set of $h$-complex numbers. Such numbers form a commutative ring that is not a field. $h$-Holomorphic functions are solutions of systems of equations of hyperbolic type, in comparison with classical holomorphic functions, which are solutions of systems of equations of elliptic type. A consequence of this is a significant difference between the properties of $h$-holomorphic functions and the classical ones. Interest in studying the properties of functions of an $h$-complex variable is associated with the need to search for new methods for solving problems in mechanics and the plane theory of relativity. The paper presents a theorem on the local invertibility of $h$-holomorphic functions, formulates the principles of preserving the domain and maximum of the norm.

Keywords: $h$-holomorphy; local invertibility; domain preservation principle; norm maximum principle; ring of $h$-complex numbers; zero divisors.

UDC: 517.58

Received: 23.04.2021
Revised: 10.01.2022
Accepted: 18.02.2022

DOI: 10.33581/2520-6508-2022-1-103-107