Template:Short description {{ safesubst:#invoke:Unsubst||date=__DATE__ |$B= Template:Ambox }} 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 indeterminateEdit
The polynomial ring Template:Math of univariate polynomials over a field Template:Math is a Template:Math-vector space, which has <math display="block">1, x, x^2, x^3, \ldots</math> as an (infinite) basis. More generally, if Template:Math is a ring then Template:Math is a free module which has the same basis.
The polynomials of degree at most Template:Math form also a vector space (or a free module in the case of a ring of coefficients), which has <math display="block">\{ 1, x, x^2, \ldots, x^{d-1}, x^d \}</math> as a basis.
The canonical form of a polynomial is its expression on this basis: <math display="block">a_0 + a_1 x + a_2 x^2 + \dots + a_d x^d,</math> or, using the shorter sigma notation: <math display="block">\sum_{i=0}^d a_ix^i.</math>
The monomial basis is naturally totally ordered, either by increasing degrees <math display="block">1 < x < x^2 < \cdots, </math> or by decreasing degrees <math display="block">1 > x > x^2 > \cdots. </math>
Several indeterminatesEdit
In the case of several indeterminates <math>x_1, \ldots, x_n,</math> a monomial is a product <math display="block">x_1^{d_1}x_2^{d_2}\cdots x_n^{d_n},</math> where the <math>d_i</math> are non-negative integers. As <math>x_i^0 = 1,</math> an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular <math> 1 = x_1^0 x_2^0\cdots x_n^0</math> is a monomial.
Similar to the case of univariate polynomials, the polynomials in <math>x_1, \ldots, x_n</math> 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 <math>d</math> form a subspace which has the monomials of degree <math>d = d_1+\cdots+d_n</math> as a basis. The dimension of this subspace is the number of monomials of degree <math>d</math>, which is <math display="block">\binom{d+n-1}{d} = \frac{n(n+1)\cdots (n+d-1)}{d!},</math> where <math display="inline">\binom{d+n-1}{d}</math> is a binomial coefficient.
The polynomials of degree at most <math>d</math> form also a subspace, which has the monomials of degree at most <math>d</math> as a basis. The number of these monomials is the dimension of this subspace, equal to <math display="block">\binom{d + n}{d}= \binom{d + n}{n}=\frac{(d+1)\cdots(d+n)}{n!}.</math>
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 <math display="block">m<n \iff mq < nq</math> and <math display="block">1 \leq m</math> for every monomial <math>m, n, q.</math>