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).