Monomial_basis Knowpia

In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists 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).

One indeterminate edit

The polynomial ring $K [x]$ of univariate polynomials over a field $K$ is a $K$ -vector space, which has

1,x,x^{2},x^{3},\ldots

as an (infinite) basis. More generally, if

K

is a ring then

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

\{1,x,x^{2},\ldots ,x^{d-1},x^{d}\}

as a basis.

The canonical form of a polynomial is its expression on this basis:

a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{d}x^{d},

or, using the shorter sigma notation:

\sum _{i=0}^{d}a_{i}x^{i}.

The monomial basis is naturally totally ordered, either by increasing degrees

1<x<x^{2}<\cdots ,

or by decreasing degrees

1>x>x^{2}>\cdots .

Several indeterminates edit

In the case of several indeterminates $x_{1},\ldots ,x_{n},$ a monomial is a product

x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{n}^{d_{n}},

where the

d_{i}

are non-negative integers. As

x_{i}^{0}=1,

an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular

1=x_{1}^{0}x_{2}^{0}\cdots x_{n}^{0}

is a monomial.

Similar to the case of univariate polynomials, the polynomials in $x_{1},\ldots ,x_{n}$ form a vector space (if the coefficients belong to a field) or a free module (if the coefficients belong to a ring), which has the set of all monomials as a basis, called the monomial basis.

The homogeneous polynomials of degree $d$ form a subspace which has the monomials of degree $d=d_{1}+\cdots +d_{n}$ as a basis. The dimension of this subspace is the number of monomials of degree $d$ , which is

{\binom {d+n-1}{d}}={\frac {n(n+1)\cdots (n+d-1)}{d!}},

where

{\textstyle {\binom {d+n-1}{d}}}

is a binomial coefficient.

The polynomials of degree at most $d$ form also a subspace, which has the monomials of degree at most $d$ as a basis. The number of these monomials is the dimension of this subspace, equal to

{\binom {d+n}{d}}={\binom {d+n}{n}}={\frac {(d+1)\cdots (d+n)}{n!}}.

In contrast to the univariate case, there is no natural total order of the monomial basis in the multivariate case. For problems which require choosing a total order, such as Gröbner basis computations, one generally chooses an admissible monomial order – that is, a total order on the set of monomials such that

m<n\iff mq<nq

and

1\leq m

for every monomial

m,n,q.

Summary

One indeterminate edit

Several indeterminates edit

See also edit