Monomial
A monomial is a product of positive integer powers of a fixed set of variables (possibly) together with a coefficient, e.g., x, 3xy^2, or -2x^2y^3z. A monomial can also be thought of as a nonzero summand of a polynomial (Becker and Weispfenning 1993, p. 191; Cox et al. 1996). A monomial with the coefficient excluded is usually called a term.
Unfortunately, in some older works, the definitions of monomial and term are sometimes reversed. Care is therefore needed in attempting to distinguish these conflicting usages.
The Wolfram Language command MonomialList [poly, {x_1, x_2, ...}] gives the list of monomials with respect to the variables x_i in the specified polynomial.
The monomials x^k and x^l are orthogonal on the unit circle |z|=1 in the complex plane (Dumitriu et al. 2004) since
The monomial functions m_lambda are defined as
where S_lambda is the set of permutations giving distinct terms in the sum and lambda is considered to be infinite (Dumitriu et al. 2004). For example.
| m_(2,1,1)=x_1^2x_2x_3. |
(3)
|
Care is needed when consulting the literature, since the distinction between terms and monomials is not always observed. For example, Dummit and Foote (2004, p. 234) define a monomial as a polynomial with only one nonzero term, without defining what is meant by "term."
See also
Binomial, Coefficient, Gröbner Basis, Monic Polynomial, Monomial Order, Polynomial, Term, Variable, TrinomialExplore with Wolfram|Alpha
References
Becker, T. and Weispfenning, V. Gröbner Bases: A Computational Approach to Commutative Algebra. New York: Springer-Verlag, 1993.Cox, D.; Little, J.; and O'Shea, D. Ideals, Varieties, and Algorithms: An Introduction to Algebraic Geometry and Commutative Algebra, 2nd ed. New York: Springer-Verlag, 1996.Dumitriu, I.; Edelman, A.; and Shuman, G. "MOPS: Multivariate Orthogonal Polynomials (Symbolically)." J. Symbolic Comput. 42, 587-620, 2007. https://doi.org/10.1016/j.jsc.200701005.Dummit, D. S. and Foote, R. M. "Examples: Polynomial Rings, Matrix Rings, and Group Rings." §7.2 in Abstract Algebra, 3rd ed. Hoboken, NJ: Wiley, pp. 233-238, 2004.Referenced on Wolfram|Alpha
MonomialCite this as:
Weisstein, Eric W. "Monomial." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Monomial.html