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

$P=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},$

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

$H(P)={\underset {i}{\max }}\,|a_{i}|\,$

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

$L(P)=\sum _{{i=0}}^{n}|a_{i}|.\,$

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

${\binom {n}{\lfloor n/2\rfloor }}^{{-1}}H(P)\leq M(P)\leq H(P){\sqrt {n+1}};$

$L(p)\leq 2^{n}M(p)\leq 2^{n}L(p);$

$H(p)\leq L(p)\leq nH(p)$

where $\scriptstyle {\binom {n}{\lfloor n/2\rfloor }}$ 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. 77. Cambridge: Cambridge University Press. p. 212. ISBN 0-521-66225-7. Zbl 0956.12001.

External links

• Polynomial height at Mathworld