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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).

One indeterminate

The polynomial ring K[x] of the univariate polynomial 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, 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

\( 1,x,x^2,\ldots \)

as a basis

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

\( a_0 + a_1 x + a_2 x^2 + \ldots + a_d x^d, \)

or, using the shorter sigma notation:

\( \sum_{i=0}^d a_ix^i. \)

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

\( 1<x<x^2<\cdots, \)

or by decreasing degrees

\( 1>x>x^2>\cdots. \)

Several indeterminates

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. Note that, as \( x_i^0=1, an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular \( 1=x_1^0x_2^0\cdots x_n^0 \) is a monomial.

Similarly 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 \( \binom{d+n-1}{d} \) denotes 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

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

Despite the univariate case, there is no natural total order of the monomial basis. For problem which require to choose a total order, such Gröbner basis computation, one generally chooses an admissible monomial order that is a total order on the set of monomials such that

\( m<n\Leftrightarrow mq<nq \)

and

\( 1\leq m \)

for every monomials m,n,q.
Notes

A polynomial can always be converted into monomial form by calculating its Taylor expansion around 0.
Examples

A polynomial in \( \Pi_4 \)

\( 1+x+3x^4 \)

See also

Horner's method
Polynomial sequence
Newton polynomial
Lagrange polynomial
Legendre polynomial
Bernstein form
Chebyshev form

Mathematics Encyclopedia

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