Norms

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Supremum norm

Let be a T2-topological space and let be the space of all bounded complex-valued continuous functions defined on . The supremum norm is the norm defined on F byThen is a commutative Banach algebra with identity.

Reznick's identity

For and polynomials in variables,where , is the Bombieri norm, andBombieri's inequality follows from this identity.

Norm

The norm of a mathematical object is a quantity that in some (possibly abstract) sense describes the length, size, or extent of the object. Norms exist for complex numbers (the complex modulus, sometimes also called the complex norm or simply "the norm"), Gaussian integers (the same as the complex modulus, but sometimes unfortunately instead defined to be the absolute square), quaternions (quaternion norm), vectors (vector norms), and matrices (matrix norms). A generalization of the absolute value known as the p-adic norm is also defined.Norms are variously denoted , , , or . In this work, single bars are used to denote the complex modulus, quaternion norm, p-adic norms, and vector norms, while the double bar is reserved for matrix norms.The term "norm" is often used without additional qualification to refer to a particular type of norm (such as a matrix norm or vector norm). Most commonly, the unqualified term "norm"..

Bombieri norm

The Bombieri -norm of a polynomial(1)is defined by(2)where is a binomial coefficient. The most remarkable feature of Bombieri's norm is that given polynomials and such that , then Bombieri's inequality(3)holds, where is the degree of , and is the degree of either or . This theorem captures the heuristic that if and have big coefficients, then so does , i.e., there can't be too much cancellation.

Bombieri inner product

For homogeneous polynomials and of degree ,

Bombieri's inequality

For homogeneous polynomials and of degree and , thenwhere is the Bombieri norm.

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