# Height of a polynomial

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

## Definition

For a polynomial P of degree n given by

${\displaystyle P=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},}$

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

${\displaystyle H(P)={\underset {i}{\max }}\,|a_{i}|\,}$

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

${\displaystyle L(P)=\sum _{i=0}^{n}|a_{i}|.\,}$

## Relation to Mahler measure

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

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

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