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

${(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor })}^{-1}H(P)\le M(P)\le H(P)\sqrt{n+1};$

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

$H(p)\le L(p)\le (n+1)H(p)$

where $(\genfrac{}{}{0ex}{}{n}{\lfloor n/2\rfloor })$ 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. 
• 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