, the Bombieri norm
, named after Enrico Bombieri
, is a norm
on homogeneous polynomials
with coefficient in
(there is also a version for univariate polynomials). This norm has many remarkable properties, the most important being listed in this article.
This norm comes from a scalar product which can be defined as follows:
we have if
- we define
In the above definition and in the rest of this article we use the following notation:
if , we write and
The most remarkable property of this norm is the Bombieri inequality:
let be two homogeneous polynomials respectively of degree and with variables, then, the following inequality holds:
||Pcdot Q||^2 leq ||P||^2 , ||Q||^2.
In fact Bombieri inequality is the left hand side of the above statement, the right and side means that Bombieri norm is a norm of algebra (giving only the left hand side is meaningless, because in this case, we can achieve the same result with any norm by multiplying the norm by a well chosen factor).
This result means that the product of two polynomials can not be arbitrarily small and this is fundamental.
Invariance by isometry
Another important property is that the Bombieri norm is invariant by composition with an
let be two homogeneous polynomials of degree with variables and let be an isometry
of (or ). Then, the we have . When this implies .
This result follows from a nice integral formulation of the scalar product:
where is the unit sphere of with its canonical measure .
be a homogeneous polynomial of degree
variables and let
. We have:
where denotes the Euclidean norm.