On the growth exponent of c-holomorphic functions with algebraic graphs

TL;DR

This paper investigates the growth behavior at infinity of c-holomorphic functions with algebraic graphs, establishing bounds, characterizations, and related theorems that connect algebraic properties with complex analytic functions.

Contribution

It introduces bounds for the growth exponent, characterizes algebraic graphs as restrictions of rational functions, and proves a Bézout-type theorem for c-holomorphic mappings with algebraic graphs.

Findings

Bound for growth exponent in terms of projective degrees

Algebraicity of graph equivalent to being a restriction of a rational function

Bézout-type theorem for generically finite c-holomorphic mappings

Abstract

This paper is the first of a series dealing with c-holomorphic functions defined on algebraic sets and having algebraic graphs. These functions may be seen as the complex counterpart of the recently introduced \textit{regulous} functions. Herein we study their growth exponent at infinity. A general result on injectivity on fibres of an analytic set together with a theorem of Tworzewski and Winiarski gives a bound for the growth exponent of a c-holomorphic function with algebraic graph in terms of the projective degrees of the sets involved. We prove also that algebricity of the graph is equivalent to the function being the restriction of a rational function (a Serre-type theorem). Then we turn to considering generically finite c-holomorphic mappings with algebraic graphs and we prove a B\'ezout-type theorem. We also study a particular case of the \L ojasiewicz inequality at infinity in this setting.