- 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:inom{n}{lfloor n/2 floor}^{-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 H(p)
where scriptstyle inom{n}{lfloor n/2 floor} 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]
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