Markov and Division Inequalities on Algebraic Sets

Abstract

A compact set \(E\subset \mathbb {C}^N\) satisfies the Markov inequality if the supremum norm on E of the gradient of a polynomial p can be estimated from above by the norm of p multiplied by a constant polynomially depending on the degree of p. This inequality is strictly related to the Bernstein approximation theorem, Schur-type estimates and the extension property of smooth functions. Additionally, the Markov inequality can be applied to the construction of polynomial grids (norming sets or admissible meshes) useful in numerical analysis. We expect such an inequality with similar consequences not only on polynomially determining compacts but also on some nowhere dense sets. The primary goal of the paper is to extend the above definition of Markov inequality to the case of compact subsets of algebraic varieties in \(\mathbb {C}^N\). Moreover, we characterize compact sets satisfying such a Markov inequality on algebraic hypersurfaces as well as on certain varieties defined by several algebraic equations. We also prove a division inequality (a Schur-type inequality) on these sets. This opens up the possibility of establishing polynomial grids on algebraic sets. We also provide examples that complete and ilustrate the results.