Substitution Property for the Ring of Continuous Rational Functions

TL;DR

This paper investigates the substitution property of the ring of continuous rational functions on real algebraic varieties, establishing conditions under which the property holds along points and Puiseux arcs, especially for non-singular varieties.

Contribution

It demonstrates that the ring R 0 (V) satisfies substitution properties along points and Puiseux arcs, providing a characterization for non-singular varieties.

Findings

R 0 (V) satisfies substitution along points.

For non-singular V, substitution also holds along Puiseux arcs.

The substitution property characterizes R 0 (V) in non-singular cases.

Abstract

We study the substitution property for the ring R 0 (V) of continuous rational functions on a real algebraic affine variety V. We show that R 0 (V) satisfies a substitution property along points; moreover, when V is non-singular, it satisfies also a substitution property along Puiseux arcs, which characterizes R 0 (V).