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Binomial ○꠹｜Definition｜1st｜20260414124507-00-⌔
Binomial (polynomial) - Wikipedia
Binomial (polynomial)
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial.1 It is the simplest kind of a sparse polynomial after the monomials.
A toric ideal is an ideal that is generated by binomials that are difference of monomials; that is, binomials whose two coefficients are 1 and −1. A toric variety is an algebraic variety defined by a toric ideal.
For every admissible monomial ordering, the minimal Gröbner basis of a toric ideal consists only of differences of monomials. (This is an immediate consequence of Buchberger’s algorithm that can produce only differences of monomials when starting with differences of monomials.
Similarly, a binomial ideal is an ideal generated by monomials and binomials (that is, the above constraint on the coefficient is released), and the minimal Gröbner basis of a binomial ideal contains only monomials and binomials. Monomials must be included in the definition of a binomial ideal, because, for example, if a binomial ideal contains and , it contains also .
Printed 2026-06-28.
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Link to original Footnotes
Weisstein, Eric W. “Binomial”. MathWorld. ↩
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