# Transcendental holomorphic maps between real algebraic manifolds in a complex space

## TL;DR

This paper presents an example of a real algebraic manifold in complex space lacking the Nash-Artin approximation Property, highlighting its implications for biholomorphic and algebraic equivalence of real algebraic manifolds.

## Contribution

It provides the first known example of a real algebraic manifold without the Nash-Artin approximation Property, using elliptic Bishop surfaces and functional equations.

## Key findings

- The example is an elliptic Bishop surface.
- The Nash-Artin approximation Property does not hold for this manifold.
- Implications for biholomorphic and algebraic equivalence problems.

## Abstract

We give an example of a real algebraic manifold embedded in a complex space that does not satisfy the Nash-Artin approximation Property. This Nash-Artin approximation Property is closely related to the problem of determining when the biholomorphic equivalence for germs of real algebraic manifolds coincides with the algebraic equivalence. This example is an elliptic Bishop surface, and its construction is based on the functional equation satisfied by the generating series of some walks restricted to the quarter plane.