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Height of a polynomial

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

Contents

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 | .

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

( 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.

References

Height of a polynomial Wikipedia


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