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Two sharp inequalities for the norm of a factor of a polynomial
Author(s) -
Boyd David W.
Publication year - 1992
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300015072
Subject(s) - monic polynomial , mathematics , unit circle , polynomial , norm (philosophy) , combinatorics , degree (music) , discrete mathematics , mathematical analysis , physics , political science , acoustics , law
Abstract Let f ( x ) be a monic polynomial of degree n with complex coefficients, which factors as f ( x ) = g ( x ) h ( x ), where g and h are monic. Let‖ f ‖be the maximum of| f ( x ) |on the unit circle. We prove that‖ g ‖ ⩽ β n ‖ f ‖ and ‖ g ‖ ‖ h ‖ ⩽ δ n ‖ f ‖ , where β = M ( P 0 ) = 1 38135 …, where P 0 is the polynomial P 0( x, y ) = 1 + x + y and δ = M ( P 1 ) = 1 79162…, where P 1 ( x, y ) = 1 + x + y ‐ xy , and M denotes Mahler's measure. Both inequalities are asymptotically sharp as n → ∞.