Height of a polynomial - Alchetron, The Free Social Encyclopedia

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

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.

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