# binomialSolve -- solving zero-dimensional binomial Ideals

## Synopsis

• 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.