Some conjectures on continuous rational maps into spheres

TL;DR

This paper explores the approximation properties of continuous rational maps into spheres, proposing a conjecture linked to classical conjectures and proving it in a special case, advancing understanding in real algebraic geometry.

Contribution

It introduces a new conjecture on continuous rational maps into spheres and proves it in a specific case, connecting it to classical conjectures in algebraic geometry.

Findings

Conjecture relates rational maps to classical transformation conjectures.

Proved the conjecture in a special case.

Derived several related approximation results.

Abstract

Recently continuous rational maps between real algebraic varieties have attracted the attention of several researchers. In this paper we continue the investigation of approximation properties of continuous rational maps with values in spheres. We propose a conjecture concerning such maps and show that it follows from certain classical conjectures involving transformation of compact smooth submanifolds of nonsingular real algebraic varieties onto subvarieties. Furthermore, we prove our conjecture in a special case and obtain several related results.