Bombieri norm

In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in ${\displaystyle \mathbb {R} }$ or ${\displaystyle \mathbb {C} }$ (there is also a version for non homogeneous univariate polynomials). This norm has many remarkable properties, the most important being listed in this article.

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

The fundamental property of this norm is the Bombieri inequality:

let ${\displaystyle P,Q}$ be two homogeneous polynomials respectively of degree ${\displaystyle d^{\circ }(P)}$ and ${\displaystyle d^{\circ }(Q)}$ with ${\displaystyle N}$ variables, then, the following inequality holds:

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