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A monomial, also called power product, is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. For example, is a monomial. The constant is a monomial, being equal to the empty product and to for any variable . If only a single variable is considered, this ...
Monomial order. In mathematics, a monomial order (sometimes called a term order or an admissible order) is a total order on the set of all ( monic) monomials in a given polynomial ring, satisfying the property of respecting multiplication, i.e., If and is any other monomial, then . Monomial orderings are most commonly used with Gröbner bases ...
Monomial ideal. In abstract algebra, a monomial ideal is an ideal generated by monomials in a multivariate polynomial ring over a field . A toric ideal is an ideal generated by differences of monomials (provided the ideal is prime ). An affine or projective algebraic variety defined by a toric ideal or a homogeneous toric ideal is an affine or ...
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 ...
Monomial representation. In the mathematical fields of representation theory and group theory, a linear representation ( rho) of a group is a monomial representation if there is a finite-index subgroup and a one-dimensional linear representation of , such that is equivalent to the induced representation . Alternatively, one may define it as a ...
The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first n power sum polynomials involves rational coefficients may depend on n. But rational coefficients are always needed to express elementary symmetric polynomials (except the constant ones, and e 1 which coincides with the first ...
In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. For a univariate polynomial, the degree of the polynomial is simply the ...
Definition. The complete homogeneous symmetric polynomial of degree k in n variables X1, ..., Xn, written hk for k = 0, 1, 2, ..., is the sum of all monomials of total degree k in the variables. Formally, The formula can also be written as: Indeed, lp is just the multiplicity of p in the sequence ik . The first few of these polynomials are.