Monomial basis

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
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: $$\sum_{i=0}^d a_ix^i.$$

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
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 $\binom{d+n-1}{d}$ 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 $$m, n, q.$$