Binomial (polynomial)
In algebra, a binomial is a polynomial that is the sum of two terms, each of which is a monomial. 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 and. 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., 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.
Definition
A binomial is a polynomial which is the sum of two monomials. A binomial in a single indeterminate can be written in the formwhere and are numbers, and and are distinct non-negative integers and 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 and may be negative.
More generally, a binomial may be written as:
Operations on simple binomials
- The binomial, the difference of two squares, can be factored as the product of two other binomials:
- The product of a pair of linear binomials and is a trinomial:
- A binomial raised to the th power, represented as can be expanded by means of the binomial theorem or, equivalently, using Pascal's triangle. For example, the square of the binomial is equal to the sum of the squares of the two terms and twice the product of the terms, that is:
- An application of the above formula for the square of a binomial is the "-formula" for generating Pythagorean triples:
- Binomials that are sums or differences of cubes can be factored into smaller-degree polynomials as follows: