Extremal products in Bombieri's norm

Abstract

Let P be a homogeneous polynomial in any number of variables, of any degree, with complex coefficients. We give a complete description of the Q's, of all degrees, such that [Q] = 1 and [PQ] is either maaimal or minimal, where $[\cdot]$ is Bombieri's norm. Far this, we introduce a matrix, built with partial derivatives of P; the quantity [PQ] appears as the l2 norm of the product of this matrix by a vector column associated with Q: thus products of polynomials are replaced by the product of a matrix by a vector, a mach simpler feature. The extreme values of [PQ] are eigenvalues of this matrix. As an application, we give an exact estimate of the norm of the differential operator P(D).