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: we have if

: $forall\; alpha\; in\; mathbb\{N\}^N$ 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 $X^alpha\; =\; Pi\_\{i=1\}^N\; X\_i^\{alpha\_i\}.$

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:

: $frac\{d^circ(P)!d^circ(Q)!\}\{(d^circ(P)+d^circ(Q))!\}||P||^2\; ,\; ||Q||^2\; leq$
|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 isometryof $mathbb\; R^N$ (or $mathbb\; C^N$). Then, the we have $langle\; Pcirc\; h|Qcirc\; h\; angle\; =\; langle\; P|Q\; angle$. When $P=Q$ this implies $||Pcirc\; h||=||P||$.

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

: $langle\; P|Q\; angle\; =\; \{d+N-1\; choose\; N-1\}\; int\_\{S^N\}\; P(Z)Q(Z),dsigma(Z)$

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

Other inequalities

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

* $|P(Z)|\; leq\; ||P||\; ,\; ||Z||\_E^d$
* $||\; abla\; P(Z)||\_E\; leq\; d\; ||P||\; ,\; ||Z||\_E^d$

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

References

* B. Beauzamy, E. Bombieri, P. Enflo and H.L. Montgomery. "Product of polynomials in many variables", "journal of number theory", pages 219--245, 1990.