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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

o3 = S

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

o4 = {{-6.30271e-72-1.04447e-71*ii, 4.87739e-72-1.27795e-71*ii,
     ------------------------------------------------------------------------
     .498229+.347753*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 => {{-6.30271e-72-1.04447e-71*ii, 4.87739e-72-1.27795e-71*ii, .498229+.347753*ii}}
                Slice => | -.212935+.272321i .17863-.837047i -1.03286-.771769i .395774+.389505i |
                WitnessDataFileName => /var/folders/j_/gx42s4z576z_vj47_ym0j5xm0000gn/T/M2-30589-0/1/witness_data
                               3      2      2       2     3      2
                Equations => {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 :