Samples points from an irreducible component of a variety using Bertini. The irreducible component needs to be in its numerical form as a WitnessSet. The method bertiniPosDimSolve can be used to generate a witness set for the component. Bertini (1) writes the witness set to a temporary file, (2) invokes Bertini's solver with option TrackType => 2, and (3 moves the hyperplanes defined in the WitnessSet W within the space until the desired points are sampled, (4) stores the output of Bertini in a temporary file, and finally (5) parses and outputs the solutions.
i1 : R = CC[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
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i3 : NV = bertiniPosDimSolve(F) o3 = NV o3 : NumericalVariety |
i4 : W = NV#1_0 --z-axis o4 = W o4 : WitnessSet |
i5 : bertiniSample(4, W)
o5 = {{0, 2.58729e-271+2.77047e-271*ii, .504375-.170678*ii},
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{2.4161e-318+4.1731e-318*ii, -2.4161e-318-4.1731e-318*ii,
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.491246-.001592*ii}, {1.60477e-272+1.72248e-272*ii,
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-4.3062e-273+4.01191e-273*ii, .500412-.168743*ii},
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{1.51709e-317+1.55709e-317*ii, 2.27564e-318+2.33564e-318*ii,
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.611325-.011571*ii}}
o5 : List
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The object bertiniSample is a method function with options.