Bernstein - Chebyshev inequality and Baran's radial extremal function on algebraic sets

We study a Bernstein-Chebyshev inequality and some Ple ' sniak type properties on polynomially determining sets and on a wide class of algebraic varieties. We show that a compact subset E of algebraic variety V satisfies a Bernstein-Chebyshev inequality if and only if a projection of E satisfies a Bernstein-Chebyshev inequality. Moreover, we give an estimate of appropriate constants. These inequalities are also studied on preimages under simple polynomial maps. Baran's radial extremal function is calculated for some compacts on algebraic sets.