Basis of polynomials consisting of monomials
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 Kx of
univariate polynomials over a field K is a K-vector space, which has
![{\displaystyle 1,x,x^{2},x^{3},\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/db592780a2a4ae86f4f12e7894d4e75381323bfc)
as an (infinite) basis. More generally, if
K is a
ring then
Kx 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
![{\displaystyle \{1,x,x^{2},\ldots ,x^{d-1},x^{d}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90854f4b1fb3ec57359b84943e6bb62e39226d31)
as a basis.
The canonical form of a polynomial is its expression on this basis:
![{\displaystyle a_{0}+a_{1}x+a_{2}x^{2}+\dots +a_{d}x^{d},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc7c21c0f2bd6e8c1ce1a5b2a6b084f13bd45b1)
or, using the shorter
sigma notation:
![{\displaystyle \sum _{i=0}^{d}a_{i}x^{i}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b3d8502696766ead421f34b276916124cd4a5dee)
The monomial basis is naturally
totally ordered, either by increasing degrees
![{\displaystyle 1<x<x^{2}<\cdots ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24df7faf69966630c48d7a221e3f7e39d2f2e9e5)
or by decreasing degrees
![{\displaystyle 1>x>x^{2}>\cdots .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5c3e322254264ba598fbb3f4feaf3acd34784720)
Several indeterminates
In the case of several indeterminates
a
monomial is a product
![{\displaystyle x_{1}^{d_{1}}x_{2}^{d_{2}}\cdots x_{n}^{d_{n}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/10523bcd61a67c90f1e39a7e9024ceee7d19177e)
where the
![{\displaystyle d_{i}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abe3154db7d4f92fb42dd1f80f52f528c6312e4a)
are non-negative
integers. As
![{\displaystyle x_{i}^{0}=1,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d0acfb116221a8844456634832bde8ab7d5898e)
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa89d7d8c54f2806020be880b31ca79bc912bc3)
is a monomial.
Similar to the case of univariate polynomials, the polynomials in
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
form a
subspace which has the monomials of degree
as a basis. The
dimension of this subspace is the number of monomials of degree
, which is
![{\displaystyle {\binom {d+n-1}{d}}={\frac {n(n+1)\cdots (n+d-1)}{d!}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6846ffdf5a622ba8068e57bc11043421ccc85fbf)
where
![{\textstyle {\binom {d+n-1}{d}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14933941212a7e20d8dbb488a10396fc42a61e7c)
is a
binomial coefficient.
The polynomials of degree at most
form also a subspace, which has the monomials of degree at most
as a basis. The number of these monomials is the dimension of this subspace, equal to
![{\displaystyle {\binom {d+n}{d}}={\binom {d+n}{n}}={\frac {(d+1)\cdots (d+n)}{n!}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e142933079c4276d559d37e9b4e3e8021f14dd)
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\iff mq<nq}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3160c07085c35fb29c69d001c20c87d902e5211)
and
![{\displaystyle 1\leq m}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f4095cdf5313929859695ca205fc1ed4ec679f7)
for every monomial
See also