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In 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 a power product or primitive monomial, [1] is a product of powers of variables with nonnegative integer exponents, or, in other words, a product of variables, possibly with repetitions. [2]
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).
In mathematics, in the area of algebra studying the character theory of finite groups, an M-group or monomial group is a finite group whose complex irreducible characters are all monomial, that is, induced from characters of degree 1. [1] In this section only finite groups are considered. A monomial group is solvable. [2]
When a monomial order has been chosen, the leading monomial is the largest u in S, the leading coefficient is the corresponding c u, and the leading term is the corresponding c u u. Head monomial/coefficient/term is sometimes used as a synonym of "leading". Some authors use "monomial" instead of "term" and "power product" instead of "monomial".
To define the monomial representation, we first need to introduce the notion of monomial space. A monomial space is a triple (,, ()) where is a finite-dimensional complex vector space, is a finite set and () is a family of one-dimensional subspaces of such that =.
For every monomial ordering, the empty set of polynomials is the unique Gröbner basis of the zero ideal. For every monomial ordering, a set of polynomials that contains a nonzero constant is a Gröbner basis of the unit ideal (the whole polynomial ring). Conversely, every Gröbner basis of the unit ideal contains a nonzero constant.
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 exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the nonzero entry must be 1, in a generalized permutation matrix the nonzero entry can be any nonzero value.
A monomial in is a product = for an n-tuple = (,, …,) of nonnegative integers. The following three conditions are equivalent for an ideal I ⊆ R {\displaystyle I\subseteq R} : I {\displaystyle I} is generated by monomials,