In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in
R
C
To start with the geometry, the Bombieri scalar product for homogeneous polynomials with N variables can be defined as follows using multi-index notation:by definition different monomials are orthogonal, so that if
\alpha ≠ \beta,
while by definition
! |
---|
In the above definition and in the rest of this article the following notation applies:
if write and and
The fundamental property of this norm is the Bombieri inequality:
let
P,Q
d\circ(P)
d\circ(Q)
N
d\circ(P)!d\circ(Q)! | |
(d\circ(P)+d\circ(Q))! |
\|P\|2\|Q\|2\leq\|P ⋅ Q\|2\leq\|P\|2\|Q\|2.
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.
Another important property is that the Bombieri norm is invariant by composition with an isometry
let
P,Q
d
N
h
RN
CN
\langleP\circh|Q\circh\rangle=\langleP|Q\rangle
P=Q
\|P\circh\|=\|P\|
This result follows from a nice integral formulation of the scalar product:
\langleP|Q\rangle={d+N-1\chooseN-1}
\int | |
SN |
P(Z)\overline{Q(Z)}d\sigma(Z)
where
SN
CN
d\sigma(Z)
Let
P
d
N
Z\inCN
|P(Z)|\leq\|P\|
d | |
\|Z\| | |
E |
\|\nablaP(Z)\|E\leqd\|P\|
d | |
\|Z\| | |
E |
where
\| ⋅ \|E
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).
. Donald Knuth. 4.6.2 Factorization of polynomials. Seminumerical algorithms. The Art of Computer Programming. 2. Third. Reading, Massachusetts. Addison-Wesley. 1997. 439–461, 678–691. 0-201-89684-2. 633878.