Height of a polynomial
Appearance
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
the height H(P) is defined to be the maximum of the magnitudes of its coefficients:
and the length L(P) is similarly defined as the sum of the magnitudes of the coefficients:
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
where 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. Zbl 1020.12001.
- Mahler, K. (1963). "On two extremum properties of polynomials". Illinois J. Math. 7: 681–701. Zbl 0117.04003.
- Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications. Vol. 77. Cambridge: Cambridge University Press. p. 212. ISBN 0-521-66225-7. Zbl 0956.12001.