Monomial basis explained

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 of univariate polynomials over a field is a -vector space, which has $$1,\; x,\; x^2,\; x^3,\; \backslash ldots$$as an (infinite) basis. More generally, if is a ring then is a free module which has the same basis.

The polynomials of degree at most form also a vector space (or a free module in the case of a ring of coefficients), which has $$\backslash$$ as a basis.

The canonical form of a polynomial is its expression on this basis:$$a\_0\; +\; a\_1\; x\; +\; a\_2\; x^2\; +\; \backslash dots\; +\; a\_d\; x^d,$$or, using the shorter sigma notation:$$\backslash sum\_^d\; a\_ix^i.$$

The monomial basis is naturally totally ordered, either by increasing degreesor by decreasing degrees$$1\; >\; x\; >\; x^2\; >\; \backslash cdots.$$

Several indeterminates

In the case of several indeterminates

x _{1,} \ldots, x _{n,} a monomial is a product$$x\_1^x\_2^\backslash cdots\; x\_n^,$$where the d _{i} are non-negative integers. As x ^{0} _{i} = 1 , an exponent equal to zero means that the corresponding indeterminate does not appear in the monomial; in particular 1 = x ^{0} _{1} x ^{0 …} _{2} x ^{0} _{n} 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+ … +d} _{n} as a basis. The dimension of this subspace is the number of monomials of degree d , which is $$\backslash binom\; =\; \backslash frac,$$where $\backslash binom$ 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$$\backslash binom=\; \backslash binom=\backslash frac.$$

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, q.

See also

• Horner's method
• Polynomial sequence
• Newton polynomial
• Lagrange polynomial
• Legendre polynomial
• Bernstein form
• Chebyshev form