# Height of a polynomial

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

For a polynomial P 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:

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

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 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.