TY - JOUR

T1 - An inequality for products of polynomials

AU - Reznick, Bruce

PY - 1993/4

Y1 - 1993/4

N2 - 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.

AB - 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.

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U2 - 10.2307/2159535

DO - 10.2307/2159535

M3 - Article

SN - 0002-9939

VL - 117

SP - 1063

EP - 1073

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 4

ER -