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Proceedings of the American Mathematical Society

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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$.

References [Enhancements On Off] (What's this?)

  • [1] B. Beauzamy, E. Bombieri, P. Enflo, and H. L. Montgomery, Products of polynomials in many variables, J. Number Theory 36 (1990), 219-245. MR 1072467 (91m:11015)
  • [2] B. Beauzamy, Products of polynomials and a priori estimates for coefficients in polynomial decompositions: a sharp result, J. Symbolic Comput. (to appear). MR 1170091 (93h:11140)
  • [3] B. Beauzamy, J.-L. Frot, and C. Millour, Massively parallel computations on many variable polynomials. II (in preparation).
  • [4] S. Helgason, Groups and geometric analysis, Academic Press, New York, 1984. MR 754767 (86c:22017)
  • [5] B. Reznick, Sums of even powers of real linear forms, Mem. Amer. Math. Soc., vol. 196, no. 463, Amer. Math. Soc., Providence, RI, 1992. MR 1096187 (93h:11043)
  • [6] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, NJ, 1971. MR 0304972 (46:4102)

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Article copyright: © Copyright 1993 American Mathematical Society

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