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

[math]\displaystyle{ P = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n , }[/math]

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

[math]\displaystyle{ H(P) = \underset{i}{\max} \,|a_i| }[/math]

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

[math]\displaystyle{ L(P) = \sum_{i=0}^n |a_i|. }[/math]

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

[math]\displaystyle{ \binom{n}{\lfloor n/2 \rfloor}^{-1} H(P) \le M(P) \le H(P) \sqrt{n+1} ; }[/math]

[math]\displaystyle{ L(p) \le 2^n M(p) \le 2^n L(p) ; }[/math]

[math]\displaystyle{ H(p) \le L(p) \le (n+1) H(p) }[/math]

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

References

• Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. Springer-Verlag. pp. 2,3,142,148. ISBN 0-387-95444-9. 
• Mahler, K. (1963). "On two extremum properties of polynomials". Illinois J. Math. 7: 681–701. 
• Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications. 77. Cambridge: Cambridge University Press. p. 212. ISBN 0-521-66225-7. 

External links

• Polynomial height at Mathworld