Basic embeddings and Hilbert's 13th problem on superpositions (in Russian)

TL;DR

This paper explores the concept of basic embeddings in the plane, related to Hilbert's 13th problem, providing descriptions, results, and open questions, aimed at both students and experienced mathematicians.

Contribution

It introduces the notion of basic subsets in the plane within the context of Hilbert's 13th problem and discusses their properties and open problems.

Findings

Descriptions of basic subsets of the plane

Solutions to Arnold's and Sternfeld's problems

Open problems on smooth basic properties

Abstract

This note is purely expository. We show how in the course of the Kolmogorov-Arnold solution of Hilbert's 13-th problem on superpositions there appeared the notion of a basic embedding. A subset K of R^2 is {\it 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 and 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.