Rational Variant of Hartogs' Theorem on Separate Holomorphicity

TL;DR

This paper proves that a partially defined function over an uncountable algebraically closed field, which is rational in each coordinate separately, must be globally rational in all variables, extending Hartogs' theorem to this algebraic setting.

Contribution

It establishes a rational variant of Hartogs' theorem for functions over uncountable algebraically closed fields, demonstrating that separate rationality implies joint rationality.

Findings

Partially defined functions rational in each coordinate are globally rational.

The result extends Hartogs' theorem to algebraic fields.

The proof applies to functions on Zariski open subsets in algebraic geometry.

Abstract

Given an uncountable algebraically closed field $K$, we proved that if partially defined function $f\colon K \times \dots \times K \dashrightarrow K$ defined on a Zariski open subset of the $n$-fold Cartesian product $K \times \dots \times K$ is rational in each coordinate whilst other coordinates are held constant, then $f$ is itself a rational function in $n$-variables.