Bombieri norm
In mathematics, the Bombieri norm, named after Enrico Bombieri, is a norm on homogeneous polynomials with coefficient in or (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 polynomials with N variables can be defined as follows using multi-index notation: by definition different monomials are orthogonal, so that if while by definition In the above definition and in the rest of this article the following notation applies: if write and and Bombieri inequality The fundamental property of this norm is the Bombieri inequality: let be two homogeneous polynomials respectively of degree and with variables, then, the following inequality holds: 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