Bombieri norm - Citizendia

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). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Enrico Bombieri (born November 26, 1940) is an Italian Mathematician, born in Milan. In Linear algebra, Functional analysis and related areas of Mathematics, a norm is a function that assigns a strictly positive length In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same 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

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 $X^\alpha = \Pi_{i=1}^N X_i^{\alpha_i}.$

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). In Mathematics, a homogeneous polynomial is a Polynomial whose terms are Monomials all having the same total degree; or are elements of the same

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 $\mathbb C^N$ ). For the Mechanical engineering and Architecture usage see Isometric projection. Then, the we have $\langle P\circ h|Q\circ h\rangle = \langle P|Q\rangle$ . When $P = Q$ this implies $||P\circ h||=||P||$ .

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

$\langle P|Q\rangle = {d+N-1 \choose N-1} \int_{S^N} P(Z)Q(Z)\,d\sigma(Z)$

where $S N$ is the unit sphere of $\mathbb C^N$ with its canonical mesure $d σ(Z)$ .

Other inequalities

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

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

where $| | . | | E$ denotes the euclidian 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. Enrico Bombieri (born November 26, 1940) is an Italian Mathematician, born in Milan. Per Enflo, born in Stockholm, Sweden in 1944 is a university professor in the Department of Mathematical Sciences at Kent State University, Ohio USA Hugh Montgomery can be Hugh Montgomery 1st Viscount of the Great Ardes Hugh Montgomery (mathematician, an American mathematician

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Contents

Bombieri scalar product for homogeneous polynomials with N variables

Bombieri inequality

Invariance by isometry

Other inequalities

References