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 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 means that a monomial is either or a power of , with a positive integer. If several variables are considered, say, then each can be given an exponent, so that any monomial is of the form with non-negative integers (taking note that any exponent makes the corresponding factor equal to ).
A monomial is a monomial in the first sense multiplied by a nonzero constant, called the coefficient of the monomial. A monomial in the first sense is a special case of a monomial in the second sense, where the coefficient is . For example, in this interpretation and are monomials (in the second example, the variables are and the coefficient is a complex number).
In the context of Laurent polynomials and Laurent series, the exponents of a monomial may be negative, and in the context of Puiseux series, the exponents may be rational numbers.
Since the word "monomial", as well as the word "polynomial", comes from the late Latin word "binomium" (binomial), by changing the prefix "bi-" (two in Latin), a monomial should theoretically be called a "mononomial". "Monomial" is a syncope by haplology of "mononomial".[1]
^American Heritage Dictionary of the English Language, 1969.
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...
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...
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...
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...
monomial group is solvable. Every supersolvable group and every solvable A-group is a monomial group. Factor groups of monomial groups are monomial,...
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...
polynomials that contain only one type of monomial, with only those copies required to obtain symmetry. Any monomial in X1, ..., Xn can be written as X1α1...
depending on the evaluation of the basis functions at the data points). The monomial basis for the vector space of analytic functions is given by { x n ∣ n...
In algebraic geometry, standard monomial theory describes the sections of a line bundle over a generalized flag variety or Schubert variety of a reductive...
requires the choice of a monomial order, that is a total order, which is compatible with the monoid structure of the monomials. Here "compatible" means...
{\displaystyle H} . To define the monomial representation, we first need to introduce the notion of monomial space. A monomial space is a triple ( V , X , (...
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field F {\displaystyle \mathbb {F} } is a linear combination...
X_{n}).} Here the "lacunary part" Placunary is defined as the sum of all monomials in P which contain only a proper subset of the n variables X1, ..., Xn...
In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following: Let A be a Noetherian local ring of Krull...
polynomial in which no variable occurs to a power of 2 or higher; that is, each monomial is a constant times a product of distinct variables. For example f(x,y...
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...
is the non-negative monomial matrices: a non-negative matrix has non-negative inverse if and only if it is a (non-negative) monomial matrix. Note that thus...
variables X1, ..., Xn, written hk for k = 0, 1, 2, ..., is the sum of all monomials of total degree k in the variables. Formally, h k ( X 1 , X 2 , … , X...
suspected that Co0 was transitive on Λ2, and indeed he found a new matrix, not monomial and not an integer matrix. Let η be the 4-by-4 matrix 1 2 ( 1 − 1 − 1 −...
multivariate polynomials with respect to a monomial order, see Gröbner basis § Leading term, coefficient and monomial. In linear algebra, a system of linear...
bi- with the Greek poly-. That is, it means a sum of many terms (many monomials). The word polynomial was first used in the 17th century. The x occurring...
elementary algebra, a trinomial is a polynomial consisting of three terms or monomials. 3 x + 5 y + 8 z {\displaystyle 3x+5y+8z} with x , y , z {\displaystyle...
each of which consists of a coefficient from R multiplied by a monomial, where each monomial is a product of finitely many finite powers of indeterminates...
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...
completeness) The algebraic degree of a function is the order of the highest order monomial in its algebraic normal form Circuit complexity attempts to classify Boolean...
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...
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