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Bertini :: bertiniPosDimSolve

bertiniPosDimSolve -- solve positive-dimensional system of equations

Synopsis

Description

The method bertiniPosDimSolve calls Bertini to find a numerical irreducible decomposition of the zero-set of F. The decomposition is returned as the NumericalVariety NV. Witness sets of NV contain approximations to solutions of the system F=0.
i1 : R = QQ[x,y,z]

o1 = R

o1 : PolynomialRing
i2 : F = {(y^2+x^2+z^2-1)*x,(y^2+x^2+z^2-1)*y}

       3      2      2       2     3      2
o2 = {x  + x*y  + x*z  - x, x y + y  + y*z  - y}

o2 : List
i3 : S = bertiniPosDimSolve F
Temporary directory for input and output files:/tmp/M2-14805-0/0

The version of Bertini you have installed on your computer was used for this run. 
Bertini is under ongoing development by D. Bates, J. Hauenstein, A. Sommese, and C. Wampler.


o3 = S

o3 : NumericalVariety
i4 : S#1_0#Points -- 1_0 chooses the first witness set in dimension 1

o4 = {{1.31165e-60+1.68487e-60*ii, -1.5376e-60-1.48332e-60*ii,
     ------------------------------------------------------------------------
     .499782-.256389*ii}}

o4 : VerticalList
Each WitnessSet is accessed by dimension and then list position.
i5 : S#1 --first specify dimension

o5 = {[dim=1,deg=1]}

o5 : List
i6 : peek oo_0 --then list position

o6 = WitnessSet{ComponentNumber => 0                                                                    }
                IsIrreducible => null
                Points => {{1.31165e-60+1.68487e-60*ii, -1.5376e-60-1.48332e-60*ii, .499782-.256389*ii}}
                Slice => | .056846+.724565i .516568+.153937i 1.49808+.948033i 2.80355+1.09854i |
                WitnessDataFileName => /tmp/M2-14805-0/0/witness_data
                                     3      2      2       2     3      2
                Equations => ideal (x  + x*y  + x*z  - x, x y + y  + y*z  - y)
In the example, we find two components, one component has dimension 1 and degree 1 and the other has dimension 2 and degree 2. We get the same results using symbolic methods.
i7 : PD=primaryDecomposition( ideal F)

             2    2    2
o7 = {ideal(x  + y  + z  - 1), ideal (y, x)}

o7 : List
i8 : dim PD_0

o8 = 2
i9 : degree PD_0

o9 = 2
i10 : dim PD_1

o10 = 1
i11 : degree PD_1

o11 = 1

Caveat

Variables must begin with a letter (lowercase or capital) and can only contain letters, numbers, underscores, and square brackets.

Ways to use bertiniPosDimSolve :