# Markov's inequality and $C^{\infty}$ functions on algebraic sets

## TL;DR

This paper extends the equivalence between Markov's and Bernstein's properties from $C^{
abla}$$
abla$ determining sets to compact subsets of algebraic varieties, deepening understanding of polynomial approximation on algebraic sets.

## Contribution

It proves an analogous equivalence result for compact subsets of algebraic varieties, generalizing known properties from smooth manifolds to algebraic sets.

## Key findings

- Markov's property is equivalent to Bernstein's property on algebraic sets.
- The result generalizes classical approximation theory to algebraic varieties.
- Provides a foundation for further study of polynomial approximation on algebraic structures.

## Abstract

It is known that for $C^{\infty}$ determining sets Markov's property is equivalent to Bernstein's property. The purpose of this paper is to prove an analogous result in the case of compact subsets of algebraic varieties.