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 $m{\text {th}}$ order differential operator, and define $||f||$ by $||f|{|^2} = f(D)\overline f$. Then $||pq|| \geqslant ||p|| \;||q||$ for all forms $p$ and $q$, regardless of degree or number of variables. Our principal result is that $||pq|| = ||p||\;||q||$ 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 $||pq|{|^2}$ in terms of the coefficients of $p$ and $q$.