## A comparative analysis of Bernstein type estimates for the derivative of multivariate polynomials

### Volume 88 / 2006

#### Abstract

We compare the yields of two methods to obtain Bernstein type pointwise estimates for the derivative of a multivariate polynomial in a domain where the polynomial is assumed to have sup norm at most 1. One method, due to Sarantopoulos, relies on inscribing ellipses in a convex domain $K$. The other, pluripotential-theoretic approach, mainly due to Baran, works for even more general sets, and uses the pluricomplex Green function (the Zaharjuta–Siciak extremal function). When the inscribed ellipse method is applied on nonsymmetric convex domains, a key role is played by the generalized Minkowski functional $\alpha(K,x)$. With the aid of this functional, our current knowledge of the best constant in the multivariate Berstein inequality is precise within a constant $\sqrt{2}$ factor. Recently L. Milev and the author derived the exact yield of the inscribed ellipse method in the case of the simplex, and a number of numerical improvements were obtained compared to the general estimates known. Here we compare the yields of this real, geometric method and the results of the complex, pluripotential-theoretical approach in the case of the simplex. We observe a few remarkable facts, comment on the existing conjectures, and formulate a number of new hypotheses.