# binomialSolve -- solving zero-dimensional binomial Ideals

## Synopsis

• Usage:
binomialSolve I
• Inputs:
• I, a pure difference binomial ideal
• Outputs:
• l, the list of solutions of I in QQ[ww]

## Description

The solutions of a pure difference binomial ideal exist in a cyclotomic field. This function will solve the ideal and construct an apropriate cyclotomic field such that the solutions are contained. If no extension is needed then the symbol that was given will remain untouched
 ```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 BinomialSolve created a cyclotomic field of order 3 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 pure difference binomial ideals.