Polynomial Norm

For a polynomial


several classes of norms are commonly defined. The l_p-norm is defined as


for p>=1, giving the special cases


Here, |P|_infty is called the polynomial height. Note that some authors (especially in the area of Diophantine analysis) use |P| as a shorthand for ||P||_infty and ||P|| as a shorthand for ||P||_2, while others (especially in the area of computational complexity) used |P| to denote the l^2-norm ||P||_2 and (Zippel 1993, p. 174).

Another class of norms is the L^p-norms, defined by


for p>=1, giving the special cases


(Borwein and Erdélyi 1995, p. 6).

See also

Bombieri Norm, Matrix Norm, Norm, Unit Circle, Vector Norm

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Borwein, P. and Erdélyi, T. "Norms on P_n." §1.1.E.3 in Polynomials and Polynomial Inequalities. New York: Springer-Verlag, pp. 6-7, 1995.Zippel, R. Effective Polynomial Computation. Boston, MA: Kluwer, 1993.

Referenced on Wolfram|Alpha

Polynomial Norm

Cite this as:

Weisstein, Eric W. "Polynomial Norm." From MathWorld--A Wolfram Web Resource.

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