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# binomialSolve -- solving zero-dimensional binomial Ideals

- Usage:
    `binomialSolve I`
- Inputs:
  - I, a unital binomial ideal
- Outputs:
  - the list of points in the zero locus of I in QQ[ww]

## Description

The solutions of a set of unital binomial equations exist in a cyclotomic field. This function will compute the variety of a unital binomial ideal and construct an appropriate cyclotomic field containing the entire variety (as a subset of the algebraic closure of QQ).

    i1 : R = QQ[x,y,z,w]

    o1 = R

    o1 : PolynomialRing

    i2 : I = ideal (x-y,y-z,z*w-1*w,w^2-x)

                                        2
    o2 = ideal (x - y, y - z, z*w - w, w  - x)

    o2 : Ideal of R

    i3 : dim I

    o3 = 0

    i4 : binomialSolve I

    o4 = {{1, 1, 1, 1}, {1, 1, 1, -1}, {0, 0, 0, 0}}

    o4 : List

    i5 : J = ideal (x^3-1,y-x,z-1,w-1)

                 3
    o5 = ideal (x  - 1, - x + y, z - 1, w - 1)

    o5 : Ideal of R

    i6 : binomialSolve J

    o6 = {{1, 1, 1, 1}, {ww , ww , 1, 1}, {- ww  - 1, - ww  - 1, 1, 1}}
                           3    3              3          3

    o6 : List

## Caveat

The current implementation can only handle unital binomial ideals.

## See also

- Cyclotomic -- cyclotomic fields

## For the programmer

The object binomialSolve is a function closure.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Binomials.m2:1637:0.