Polynomial Norm

Polynomial norms measure the size of a polynomial. 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) use to denote the -norm (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).

The integral formula extends to a quasi-norm for , and its limiting value as is the Mahler measure. Zhang (2026) proved that, for a polynomial of polynomial degree with all its roots on the unit circle, the ratio is nondecreasing for , with these endpoint conventions. This is the Baernstein quasi-norm monotonicity conjecture.