Polynomial inequalities on certain algebraic hypersurfaces

We prove that any Markov set in CN satisfies a Schur type inequality for polynomials and we give a generalization for polynomial matrices. As a consequence, we obtain polynomial inequalities on compact subsets of algebraic hypersurfaces of the form V={zN+1k=s(z1,…,zN)}⊂CN+1, where s is a non constant polynomial of N variables. We also give a condition equivalent to the Markov inequality on compact subsets of V.