Lesson plan
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. (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 .
Article summary
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. (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 .
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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