Monomial order information

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...

mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered: A monomial, also called...

number of variables, every monomial order is thus the restriction to N n {\displaystyle \mathbb {N} ^{n}} of a monomial order of Z n {\displaystyle \mathbb...

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...

monomial order is generally fixed. In this case, a polynomial may be said to be monic, if it has 1 as its leading coefficient (for the monomial order)...

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...

extends to monomials; thus, sin 3x = sin(3x) and even sin 1/2xy = sin(xy/2), but sin x + y = sin(x) + y, because x + y is not a monomial. However, this...

multivariate polynomials with respect to a monomial order, see Gröbner basis § Leading term, coefficient and monomial. In linear algebra, a system of linear...

terms of each factor (this is true whenever one uses a monomial order, like the lexicographic order used here), and the leading term of the factor ei (X1...

Linearly ordered group Monomial order Weak order of permutations Bruhat order on a Coxeter group Incidence algebra Monotonic Pointwise order of functions Galois...

This generalization depends heavily on the notion of a monomial order. The choice of an ordering on the variables is already implicit in Gaussian elimination...

that can be expressed as the tropical sum of a finite number of monomial terms. A monomial term is a tropical product (and/or quotient) of a constant and...

}p\geq \theta _{\mu }q.} Each derivative has an integer tuple, and a monomial order ranks the derivative by ranking the derivative's integer tuple. The...

polynomial) in any of its monomials; the multiplicative order, that is, the number of times the polynomial is divisible by some value; the order of the polynomial...

In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is...

that is not monomial: since the abelianization of this group has order three, its irreducible characters of degree two are not monomial. Isaacs (1994)...

is a leading monomial of some element of the Gröbner basis which is a pure power of this variable. For this test, the best monomial order (that is the...

extension of the basis field), if and only if the Gröbner basis for any monomial ordering is reduced to {1}. By means of the Hilbert series one may compute...

{\displaystyle I} for any admissible monomial ordering (the initial ideal of I {\displaystyle I} is the set of all leading monomials of elements of I {\displaystyle...

symmetric functions, for example as in xy + yz + zx where writing the final monomial as xz would distract from the pattern. A substantial use of cyclic orders...

bases including her discovery of the FGLM algorithm for changing monomial orderings in Gröbner bases, and for her development of the components of the...

respect to a monomial order and a second monomial order. As its output, it returns a Gröbner basis of the ideal with respect to the second ordering. The algorithm...

series. Let α be a multi-index for a power series f(x1, x2, …, xn). The order of the power series f is defined to be the least value r {\displaystyle...

of I for a monomial ordering refining the total degree partial ordering and G the (homogeneous) ideal generated by the leading monomials of the elements...

in J (usual sum of vectors). In particular, the product of two monomials is a monomial whose exponent vector is the sum of the exponent vectors of the...