Examples of "monomials"
introduced monomials associated to standard Young tableaux.
For example, consider the monomials formula_10, formula_11, formula_12, and formula_13; the monomial orders above would order these four monomials as follows:
All operations related to Gröbner bases require the choice of a total order on the monomials, with the following properties of compatibility with multiplication. For all monomials , , ,
that is sums of "divided power monomials" of the form
The number of monomials of degree at most in variables is formula_14 This follows from the one-to-one correspondence between the monomials of degree in variables and the monomials of degree at most in variables, which consists in substituting by 1 the extra variable.
monomials coming from odd admissible sequences composed of k blocks.
Monomials in "n" variables define homogeneous functions . For example,
the permutation representations and the associated monomials are connected by
The simplest symmetric function containing all of these monomials is
by definition different monomials are orthogonal, so that
In mathematics, a moment matrix is a special symmetric square matrix whose rows and columns are indexed by monomials. The entries of the matrix depend on the product of the indexing monomials only (cf. Hankel matrices.)
In mathematics the monomial basis of a polynomial ring is its basis (as vector space or free module over the field or ring of coefficients) that consists in the set 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).
The polynomials of degree at most formula_13 form also a subspace, which has the monomials of degree at most formula_13 as a basis. The number of these monomials is the dimension of this subspace, equal to
The homogeneous polynomials of degree formula_13 form a subspace which has the monomials of degree formula_14 as a basis. The dimension of this subspace is the number of monomials of degree formula_13, which is
The Hilbert series is a compact way to express the number of monomials of a given degree: the number of monomials of degree in variables is the coefficient of degree of the formal power series expansion of
are called the monomials associated with formula_107 with respect to formula_6, resp. formula_12.
This is sometimes also called the homogeneity operator, because its eigenfunctions are the monomials in "z":
Finally the monomials formula_23, formula_24 are polynomials in formula_21 with positive coefficients
we have the following relations between the monomials and permutations corresponding to an element formula_40:
The theorem, in a special case, states that a necessary and sufficient condition for the monomials