Definitions

# Bombieri norm

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.

## Bombieri scalar product for homogeneous polynomials with N variables

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

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

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

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

## Bombieri inequality

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

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

$frac\left\{d^circ\left(P\right)!d^circ\left(Q\right)!\right\}\left\{\left(d^circ\left(P\right)+d^circ\left(Q\right)\right)!\right\}||P||^2 , ||Q||^2 leq$
||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 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 $langle Pcirc h|Qcirc hrangle = langle P|Qrangle$. When $P=Q$ this implies $||Pcirc h||=||P||$.

This result follows from a nice integral formulation of the scalar product:

$langle P|Qrangle = \left\{d+N-1 choose N-1\right\} int_\left\{S^N\right\} P\left(Z\right)Q\left(Z\right),dsigma\left(Z\right)$

where $S^N$ is the unit sphere of $mathbb C^N$ with its canonical measure $dsigma\left(Z\right)$.

## Other inequalities

Let $P$ be a homogeneous polynomial of degree $d$ with $N$ variables and let $Z in mathbb C^N$. We have:

• $|P\left(Z\right)| leq ||P|| , ||Z||_E^d$
• $||nabla P\left(Z\right)||_E leq d ||P|| , ||Z||_E^d$

where $||.||_E$ denotes the Euclidean norm.

## References

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