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 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:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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 Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \scriptstyle \binom{n}{\lfloor n/2 \rfloor}

is the binomial coefficient. 

External links

• Polynomial height at Mathworld