An Inequality for Products of Polynomials

Abstract

Beauzamy, Bombieri, Enflo, and Montgomery recently established an inequality for the coefficients of products of homogeneous polynomials in several variables with complex coefficients (forms). We give this inequality an alternative interpretation: let f be a form of degree m, let f(D) denote the associated mth order differential operator, and define IIf I by If 112 = f(D)f. Then lIpqll > IIPII liqll for all forms p and q, regardless of degree or number of variables. Our principal result is that flpql1 = IIPII lIqI if and only if, after a unitary change of variables, p and q are forms in disjoint sets of variables. This is achieved via an explicit formula for flpq112 in terms of the coefficients of p and q.