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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 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 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 ...
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).
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 ...
The monomial symmetric functions m α. Suppose α = (α 1,α 2,...) is a sequence of non-negative integers, only finitely many of which are non-zero. Then we can consider the monomial defined by α: X α = X 1 α 1 X 2 α 2 X 3 α 3.... Then m α is the symmetric function determined by X α, i.e. the sum of all monomials obtained from X α by ...
For example f(x,y,z) = 3xy + 2.5 y - 7z is a multilinear polynomial of degree 2 (because of the monomial 3xy) whereas f(x,y,z) = x² +4y is not. The degree of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial.