Piecewise Weierstrass preparation and division for o-minimal holomorphic functions

TL;DR

This paper establishes piecewise Weierstrass preparation and division theorems for definable holomorphic functions within o-minimal structures, extending classical complex analysis results to a definable setting with applications to Nullstellensatz.

Contribution

It introduces the first piecewise versions of Weierstrass theorems for o-minimal holomorphic functions, including applications to algebraic geometry.

Findings

Proves piecewise Weierstrass preparation theorem for definable holomorphic functions.

Establishes piecewise Weierstrass division theorem in o-minimal structures.

Derives a definable global Nullstellensatz for principal ideals.

Abstract

Given an o-minimal structure expanding the field of reals, we show a piecewise Weierstrass preparation theorem and a piecewise Weierstrass division theorem for definable holomorphic functions. In the semialgebraic setting and for the structure of globally subanalytic sets and functions we obtain the corresponding results for definable real analytic functions. As an application we show a definable global Nullstellensatz for principal ideals.