Height of a polynomial

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In mathematics, the height and length of a polynomial P with complex coefficients are measures of its "size".

For a polynomial P of degree n given by

P = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n ,

the height H(P) is defined to be the maximum of the magnitudes of its coefficients:

H(P) = \underset{i}{\max} \,|a_i| \,

and the length L(P) is similarly defined as the sum of the magnitudes of the coefficients:

L(P) = \sum_{i=0}^n |a_i|.\,

The Mahler measure M(P) of P is also a measure of the size of P. The three functions H(P), L(P) and M(P) are related by the inequalities

\binom{n}{\lfloor n/2 \rfloor}^{-1} H(P) \le M(P) \le H(P) \sqrt{n+1} ;
L(p) \le 2^n M(p) \le 2^n L(p) ;
H(p) \le L(p) \le n H(p)

where \scriptstyle \binom{n}{\lfloor n/2 \rfloor} is the binomial coefficient.


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