An inequality for products of polynomials

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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 /be a form of degree m, let f(D) denote the associated mth order differential operator, and define ∥f∥ by ∥f∥2 = f(D)f. Then ∥pq∥ ≥ ∥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.

Original languageEnglish (US)
Pages (from-to)1063-1073
Number of pages11
JournalProceedings of the American Mathematical Society
Issue number4
StatePublished - Apr 1993


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