Height of a polynomial

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

For a polynomial "P" 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:

:$L(P)\; =\; sum\_\{i=0\}^n\; |a\_i|.,$

For a complex polynomial "P" of degree "n", the height "H"("P"), length "L"("P") and Mahler measure "M"("P") are related by the double inequalities

:

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

:$H(p)\; le\; L(p)\; le\; n\; H(p)$

where is the binomial coefficient.

References

*cite book | author=Peter Borwein | authorlink=Peter Borwein | title=Computational Excursions in Analysis and Number Theory | series=CMS Books in Mathematics | publisher=Springer-Verlag | year=2002 | isbn=0-387-95444-9 | pages=2,3,142,148
*

External links

* [http://mathworld.wolfram.com/PolynomialHeight.html Polynomial height at Mathworld]