Monomial basis - Alchetron, The Free Social Encyclopedia

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 , …

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 , …

as a basis

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

a 0 + a 1 x + a 2 x 2 + … + a d x d ,

or, using the shorter sigma notation:

∑ i = 0 d a i x i .

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

1 < x < x 2 < ⋯ ,

or by decreasing degrees

1 > x > x 2 > ⋯ .

Several indeterminates

In the case of several indeterminates x 1 , … , x n , a monomial is a product

x 1 d 1 x 2 d 2 ⋯ 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 0 x 2 0 ⋯ x n 0 is a monomial.

Similar to the case of univariate polynomials, the polynomials in x 1 , … , 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 + ⋯ + d n as a basis. The dimension of this subspace is the number of monomials of degree d , which is

( d + n − 1 d ) = n ( n + 1 ) ⋯ ( n + d − 1 ) d ! ,

where ( 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

( d + n d ) = ( d + n n ) = ( d + 1 ) ⋯ ( 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 ⇔ m q < n q

and

1 ≤ m

for every monomials m , n , q .

A polynomial can always be converted into monomial form by calculating its Taylor expansion around 0. For example, a polynomial in Π 4 :

1 + x + 3 x 4

References

Monomial basis Wikipedia

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Contents

One indeterminate

Several indeterminates

References