On some regularity condition

TL;DR

This paper proves that if a function over an uncountable field of characteristic zero is regular on every affine plane, then it is globally regular, establishing a key regularity condition.

Contribution

It introduces a new regularity criterion for functions based on their behavior on affine planes over uncountable fields.

Findings

Functions regular on all affine planes are globally regular

The result applies to fields of characteristic zero

Provides a new perspective on regularity conditions in algebraic geometry

Abstract

Let $\mathbb{K}$ be an uncountable field of characteristic zero and let $f$ be a function from $\mathbb{K}^n$ to $\mathbb{K}$. We show that if the restriction of $f$ to every affine plane $L\subset\mathbb{K}^n$ is regular, then $f$ is a regular function.