Bombieri norm

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

1 Bombieri scalar product for homogeneous polynomials with N variables
2 Bombieri inequality
3 Invariance by isometry
4 Other inequalities
5 References

[edit] Bombieri scalar product for homogeneous polynomials with N variables

This norm comes from a scalar product which can be defined as follows: $\forall \alpha,\beta \in \mathbb{N}^N$ we have $\langle X^\alpha | X^\beta \rangle = 0$ if $\alpha \neq \beta$

$\forall \alpha \in \mathbb{N}^N$ we define $||X^\alpha||^2 = \frac{|\alpha|!}{\alpha!}.$

In the above definition and in the rest of this article we use the following notation:

if $\alpha = (\alpha_1,\dots,\alpha_N) \in \mathbb{N}^N$ , we write $|\alpha| = \Sigma_{i=1}^N \alpha_i$ and $\alpha! = \Pi_{i=1}^N (\alpha_i!)$ and $X^\alpha = \Pi_{i=1}^N X_i^{\alpha_i}.$

[edit] Bombieri inequality

The most remarkable property of this norm is the Bombieri inequality:

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

$\frac{d^\circ(P)!d^\circ(Q)!}{(d^\circ(P)+d^\circ(Q))!}||P||^2 \, ||Q||^2 \leq ||P\cdot Q||^2 \leq ||P||^2 \, ||Q||^2.$

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.

[edit] Invariance by isometry

Another important property is that the Bombieri norm is invariant by composition with an isometry:

let $P, Q$ be two homogeneous polynomials of degree $d$ with $N$ variables and let $h$ be an isometry of $\mathbb R^N$ (or $\mathbb C^N$ ). Then, the we have $\langle P\circ h|Q\circ h\rangle = \langle P|Q\rangle$ . When $P = Q$ this implies $||P\circ h||=||P||$ .