A REMEZ-TYPE THEOREM FOR HOMOGENEOUS POLYNOMIALS

Remez-type inequalities provide upper bounds for the uniform norms of polynomials $p$ on given compact sets $K$, provided that $|p(x)|\leq1$ for every $x\in K\setminus E$, where $E$ is a subset of $K$ of small measure. In this paper we prove sharp Remez-type inequalities for homogeneous polynomials on star-like surfaces in $\mathbb{R}^d$. In particular, this covers the case of spherical polynomials (when $d=2$ we deduce a result of Erdélyi for univariate trigonometric polynomials).