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| In [[mathematics]], the '''height''' and '''length''' of a polynomial ''P'' with [[complex numbers|complex]] coefficients are measures of its "size".
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| For a [[polynomial]] ''P'' given by
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| :<math>P = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n , </math>
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| the '''height''' ''H''(''P'') is defined to be the maximum of the magnitudes of its coefficients:
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| :<math>H(P) = \underset{i}{\max} \,|a_i| \,</math>
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| and the '''length''' ''L''(''P'') is similarly defined as the sum of the magnitudes of the coefficients:
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| :<math>L(P) = \sum_{i=0}^n |a_i|.\,</math>
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| For a complex polynomial ''P'' of degree ''n'', the height ''H''(''P''), length ''L''(''P'') and [[Mahler measure]] ''M''(''P'') are related by the double [[inequality (mathematics)|inequalities]]
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| :<math>\binom{n}{\lfloor n/2 \rfloor}^{-1} H(P) \le M(P) \le H(P) \sqrt{n+1} ; </math> | |
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| :<math>L(p) \le 2^n M(p) \le 2^n L(p) ; </math>
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| :<math>H(p) \le L(p) \le n H(p) </math>
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| where <math>\scriptstyle \binom{n}{\lfloor n/2 \rfloor}</math> is the [[binomial coefficient]].
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| ==References==
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| *{{cite book | author=Peter Borwein | authorlink=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | series=CMS Books in Mathematics | publisher=[[Springer-Verlag]] | year=2002 | isbn=0-387-95444-9 | pages=2,3,142,148 }}
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| * {{cite journal | author=K. Mahler | authorlink=Kurt Mahler | title=On two extremum properties of polynomials | journal=Illinois J. Math. | volume=7 | pages=681–701 | year= 1963 }}
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| ==External links==
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| * [http://mathworld.wolfram.com/PolynomialHeight.html Polynomial height at Mathworld]
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| [[Category:Number theory]]
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| [[Category:Polynomials]]
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| {{numtheory-stub}}
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| {{mathanalysis-stub}}
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