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In [[mathematics]], the '''Bombieri norm''', named after [[Enrico Bombieri]], is a [[Norm (mathematics)|norm]] on [[homogeneous polynomial]]s with coefficient in <math>\mathbb R</math> or <math>\mathbb C</math> (there is also a version for non homogeneous univariate polynomials). This norm has many remarkable properties, the most important being listed in this article.

==Bombieri scalar product for homogeneous polynomials==

To start with the geometry, the ''Bombieri scalar product'' for [[homogeneous polynomial]]s with ''N'' variables can be defined as follows using [[multi-index notation]]:

:<math>\forall \alpha,\beta \in \mathbb{N}^N</math>

by definition different monomials are orthogonal, so that

:<math>\langle X^\alpha | X^\beta \rangle = 0</math> if <math>\alpha \neq \beta,</math>

while

:<math>\forall \alpha \in \mathbb{N}^N</math>

by definition

:<math>\|X^\alpha\|^2 = \frac{\alpha!}{|\alpha|!}.</math>

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

if

:<math>\alpha = (\alpha_1,\dots,\alpha_N) \in \mathbb{N}^N,</math>

write

:<math>|\alpha| = \Sigma_{i=1}^N \alpha_i</math>

and

:<math>\alpha! = \Pi_{i=1}^N (\alpha_i!)</math>

and

:<math>X^\alpha = \Pi_{i=1}^N X_i^{\alpha_i}.</math>

==Bombieri inequality==
The fundamental property of this norm is the Bombieri inequality:

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

:<math>\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.</math>

Here the Bombieri inequality is the left hand side of the above statement, while the right side means that the Bombieri norm is an [[algebra norm]]. Giving the left hand side is meaningless without that constraint, because in this case, we can achieve the same result with any norm by multiplying the norm by a well chosen factor.

This multiplicative inequality implies that the product of two polynomials is bounded from below by a quantity that depends on the multiplicand polynomials. Thus, this product can not be arbitrarily small. This multiplicative inequality is useful in metric [[algebraic geometry]] and [[number theory]].

==Invariance by isometry==
Another important property is that the Bombieri norm is invariant by composition with an
[[isometry]]:

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

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

: <math>\langle P|Q\rangle = {d+N-1 \choose N-1} \int_{S^N} P(Z)\overline{Q(Z)}\,d\sigma(Z)</math>

where <math>S^N</math> is the unit sphere of <math>\mathbb C^N</math> with its canonical measure <math>d\sigma(Z)</math>.

==Other inequalities==
Let <math>P</math> be a homogeneous polynomial of degree <math>d</math> with <math>N</math> variables and let <math>Z \in \mathbb C^N</math>. We have:

* <math>|P(Z)| \leq \|P\| \, \|Z\|_E^d</math>
* <math>\|\nabla P(Z)\|_E \leq d \|P\| \, \|Z\|_E^d</math>

where <math>\|\cdot\|_E</math> denotes the Euclidean norm.

The Bombieri norm is useful in polynomial factorization, where it has some advantages over the [[Mahler measure]], according to Knuth (Exercises 20-21, pages 457-458 and 682-684).

==See also==
*[[Grassmann manifold]]
*[[Hardy space]]
*[[Homogeneous polynomial]]
*[[Plücker embedding]]

==References==
* {{cite journal|last1=Beauzamy|first1=Bernard|last2=Bombieri|first2=Enrico|authorlink2=Enrico Bombieri|last3=Enflo|first3=Per|authorlink3=Per Enflo |last4=Montgomery|first4=Hugh L.|authorlink4=Hugh Montgomery (mathematician)|title=Products of polynomials in many variables|journal=[[Journal of Number Theory]]|volume=36|year=1990|issue=2|pages=219–245| doi = 10.1016/0022-314X(90)90075-3 | url2 = http://deepblue.lib.umich.edu/handle/2027.42/28840|url=http://deepblue.lib.umich.edu/bitstream/2027.42/28840/1/0000675.pdf|mr=1072467 }}
* {{cite journal|title=Quantitative estimates for polynomials in one or several variables: From analysis and number theory to symbolic and massively parallel computation|last1=Beauzamy|first1=Bernard|last2=Enflo|first2=Per|authorlink2=Per Enflo |last3=Wang|first3=Paul
|journal=Mathematics Magazine
|volume=67
|issue=4
|date=October 1994
|pages=243–257
|url=http://www.scmsa.eu/archives/ART_quantitative_estimates_1992.pdf|jstor=2690843|mr=1300564|doi=10.2307/2690843|unused_data=}}
*{{cite book
|first1=Enrico|last1=Bombieri|authorlink1=Enrico Bombieri|last2=Gubler|first2=Walter
|title=Heights in Diophantine geometry
|publisher=Cambridge U. P.
|year=2006|
isbn=0-521-84615-3
|mr=2216774}}
*{{cite book
|authorlink=Donald Knuth|last=Knuth|first=Donald E.
|chapter=[[The_Art_of_Computer_Programming#Chapters|4.6.2 Factorization of polynomials]]
|title=Seminumerical algorithms
|series = ''[[The Art of Computer Programming]]''
|volume=2
|edition=Third
|location=Reading, Massachusetts
|publisher=Addison-Wesley
|year=1997
|pages=439–461, 678–691
|isbn=0-201-89684-2|mr=633878}}

[[Category:Norms (mathematics)]]
[[Category:Analytic number theory]]
[[Category:Polynomials]]
[[Category:Homogeneous polynomials]]
[[Category:Complex analysis]]
[[Category:Several complex variables]]

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