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

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.