Extrema of Multivariate Polynomials, Their Gradients and Directional Derivatives

Abstract

O. D. Kellogg in [K28] established a connection between the supremum norms of a homogeneous polynomial and its gradient. We completed this result with a characterization of extrema in the bivariate case in [H94a] and announced the extension for multivariate homogeneous polynomials. The extension is presented in this paper. The generalization of the completed result to the case of arbitrary multivariate polynomials is also given here. The bivariate case of this contains, as special cases, the Bernstein and Markoff inequalities. Next, a well-known equality, involving suprema over directions of derivatives, is discussed. This relation turned out to be a dual to the above result in the homogeneous case (see [H94b]). On this basis, the sets of directions of suprema of the equality are characterized.