Bombieri norm

In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in formula_1 or formula_2 (there is also a version for non homogeneous univariate polynomials). This norm has many remarkable properties, the most important being listed in this article.
Bombieri scalar product for homogeneous polynomials.
To start with the geometry, the "Bombieri scalar product" for homogeneous polynomials with "N" variables can be defined as follows using multi-index notation:
by definition different monomials are orthogonal, so that
while
by definition
In the above definition and in the rest of this article the following notation applies:
if
write
and
and
Bombieri inequality.
The fundamental property of this norm is the Bombieri inequality:
let formula_12 be two homogeneous polynomials respectively of degree formula_13 and formula_14 with formula_15 variables, then, the following inequality holds:
Here the Bombieri inequality is the left hand side of the above statement, while the right side means that the Bombieri norm is an algebra norm. Giving the left hand side is meaningless without that constraint, because in this case, we can achieve the same result with any norm by multiplying the norm by a well chosen factor.
This multiplicative inequality implies that the product of two polynomials is bounded from below by a quantity that depends on the multiplicand polynomials. Thus, this product can not be arbitrarily small. This multiplicative inequality is useful in metric algebraic geometry and number theory.
Invariance by isometry.
Another important property is that the Bombieri norm is invariant by composition with an 
isometry:
let formula_12 be two homogeneous polynomials of degree formula_18 with formula_15 variables and let formula_20 be an isometry
of formula_21 (or formula_22). Then, the we have formula_23. When formula_24 this implies formula_25.
This result follows from a nice integral formulation of the scalar product:
where formula_27 is the unit sphere of formula_22 with its canonical measure formula_29.
Other inequalities.
Let formula_30 be a homogeneous polynomial of degree formula_18 with formula_15 variables and let formula_33. We have:
where formula_36 denotes the Euclidean norm.
The Bombieri norm is useful in polynomial factorization, where it has some advantages over the Mahler measure, according to Knuth (Exercises 20-21, pages 457-458 and 682-684).