Monomial basis

Several indeterminates

In the case of several indeterminates ${\displaystyle x_1,\dots ,x_n,}$ a monomial is a product $${\displaystyle x_1^{d_1}{x}_2^{d_2}\cdots x_n^{d_n},}$$ where the ${\displaystyle d_i}$ are non-negative integers. As ${\displaystyle x_i^0=1,}$ an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular ${\displaystyle 1=x_1^0{x}_2^0\cdots x_n^0}$ is a monomial.

Similar to the case of univariate polynomials, the polynomials in ${\displaystyle x_1,\dots ,x_n}$ 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 ${\displaystyle d}$ form a subspace which has the monomials of degree ${\displaystyle d=d_1+\cdots +d_n}$ as a basis. The dimension of this subspace is the number of monomials of degree ${\displaystyle d}$, which is $${\displaystyle (\genfrac{}{}{0ex}{}{d+n-1}{d})=\frac{n(n+1)\cdots (n+d-1)}{d!},}$$ where $(\genfrac{}{}{0ex}{}{d+n-1}{d})$ is a binomial coefficient.

The polynomials of degree at most ${\displaystyle d}$ form also a subspace, which has the monomials of degree at most ${\displaystyle d}$ as a basis. The number of these monomials is the dimension of this subspace, equal to $${\displaystyle (\genfrac{}{}{0ex}{}{d+n}{d})=(\genfrac{}{}{0ex}{}{d+n}{n})=\frac{(d+1)\cdots (d+n)}{n!}.}$$

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 $${\displaystyle m<n⟺mq<nq}$$ and $${\displaystyle 1\le m}$$ for every monomial ${\displaystyle m,n,q.}$