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Binomials :: binomialSolve

binomialSolve -- solving zero-dimensional binomial Ideals

Synopsis

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.

See also