Monomial basis
In mathematics the monomial basis of a polynomial ring is its basis (as vector space or free module over the field or ring of coefficients) that consists in the set of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial).
The polynomial ring K[x] of the univariate polynomial over a field K is a K-vector space, which has
as an (infinite) basis. More generally, if K is a ring, K[x] is a free module, which has the same basis.
The polynomials of degree at most d form also a vector space (or a free module in the case of a ring of coefficients), which has
as a basis
The canonical form of a polynomial is its expression on this basis:
or, using the shorter sigma notation:
The monomial basis is naturally totally ordered, either by increasing degrees
or by decreasing degrees
In the case of several indeterminates a monomial is a product
...
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