Binomial (polynomial)

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 ⁠${\displaystyle y-x}$⁠ and ⁠${\displaystyle y-2x}$⁠, it contains also ⁠${\displaystyle (y-x)-(y-2x)=x}$⁠.

Definition

A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate (also known as a univariate binomial) can be written in the form

${\displaystyle a{x}^m-b{x}^n,}$

where a and b are numbers, and m and n are distinct non-negative integers and x is a symbol which is called an indeterminate or, for historical reasons, a variable. In the context of Laurent polynomials, a Laurent binomial, often simply called a binomial, is similarly defined, but the exponents m and n may be negative.

More generally, a binomial may be written as:

${\displaystyle a{x}_1^{n_1}\cdots x_i^{n_i}-b{x}_1^{m_1}\cdots x_i^{m_i}}$