Polynomial Norm

For a polynomial

(1)

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

(2)

for , giving the special cases

			(3)
			(4)
			(5)

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

Another class of norms is the -norms, defined by

(6)

for , giving the special cases

			(7)
			(8)
			(9)

(Borwein and Erdélyi 1995, p. 6).

REFERENCES:

Borwein, P. and Erdélyi, T. "Norms on ." §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.