Basic embeddings and Hilbert's 13th problem

TL;DR

This paper explores the concept of basic embeddings in the plane, providing characterizations, proofs, and discussing open problems related to smooth versions, within an expository framework accessible to undergraduates.

Contribution

It offers a comprehensive description and proof of basic subsets of the plane and their embeddability, addressing classical problems and open questions in the field.

Findings

Characterization of basic subsets of the plane

Solutions to Arnold's and Sternfeld's problems on embeddability

Open problems on smooth basic embeddings

Abstract

This note is purely expository. In the course of the Kolmogorov-Arnold solution of Hilbert's 13th problem on superpositions there appeared the notion of basic embedding. A subset K of R^2 is basic if for each continuous function f:K->R there exist continuous functions g,h:R->R such that f(x,y)=g(x)+h(y) for each point (x,y) in K. We present descriptions of basic subsets of the plane (with a proof) and description of graphs basically embeddable into the plane (solutions of Arnold's and Sternfeld's problems). We present some results and open problems on the smooth version of the property of being basic. This note is accessible to undergraduates and could be an interesting easy reading for mature mathematicians. The two sections can be read independently on each other.