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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An inequality for products of polynomials

Author: Bruce Reznick
Journal: Proc. Amer. Math. Soc. 117 (1993), 1063-1073
MSC: Primary 11E76; Secondary 26C05
MathSciNet review: 1119265
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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 $ \vert\vert f\vert\vert$ by $ \vert\vert f\vert{\vert^2} = f(D)\overline f $. Then $ \vert\vert pq\vert\vert \geqslant \vert\vert p\vert\vert\,\;\vert\vert q\vert\vert$ for all forms $ p$ and $ q$, regardless of degree or number of variables. Our principal result is that $ \vert\vert pq\vert\vert = \vert\vert p\vert\vert\;\vert\vert q\vert\vert$ 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 $ \vert\vert pq\vert{\vert^2}$ in terms of the coefficients of $ p$ and $ q$.

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Article copyright: © Copyright 1993 American Mathematical Society