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
+
⋯
+
a
n
x
n
,
the height H(P) is defined to be the maximum of the magnitudes of its coefficients:
H
(
P
)
=
max
i

a
i

and the length L(P) is similarly defined as the sum of the magnitudes of the coefficients:
L
(
P
)
=
∑
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
(
n
⌊
n
/
2
⌋
)
−
1
H
(
P
)
≤
M
(
P
)
≤
H
(
P
)
n
+
1
;
L
(
p
)
≤
2
n
M
(
p
)
≤
2
n
L
(
p
)
;
H
(
p
)
≤
L
(
p
)
≤
n
H
(
p
)
where
(
n
⌊
n
/
2
⌋
)
is the binomial coefficient.