bertiniSample -- a main method to sample points from an irreducible component of a variety

Synopsis

Usage:

V = bertiniSample (n, W)
Inputs:
- n, an integer, an integer specifying the number of desired sample points
- W, a witness set, a witness set for an irreducible component
Optional inputs:
- BertiniInputConfiguration (missing documentation) => ..., default value {},
- IsProjective => ..., default value -1, optional argument to specify whether to use homogeneous coordinates
- Verbose => ..., default value false, Option to silence additional output
Outputs:
- L, a list, a list of sample points

Description

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

i3 : NV = bertiniPosDimSolve(F)

o3 = NV

o3 : NumericalVariety

i4 : W = NV#1_0 --z-axis

o4 = W

o4 : WitnessSet

i5 : bertiniSample(4, W)

o5 = {{-3.98895e-21-2.66903e-21*ii, -1.09049e-20+2.39057e-21*ii,
     ------------------------------------------------------------------------
     .335497+.0877674*ii}, {-2.38851e-21+7.43066e-21*ii,
     ------------------------------------------------------------------------
     7.04885e-21-3.31435e-21*ii, .235164+.0679316*ii},
     ------------------------------------------------------------------------
     {-7.16418e-21+3.40793e-21*ii, 8.68665e-21+1.35454e-20*ii,
     ------------------------------------------------------------------------
     .376396+.160188*ii}, {7.24583e-21+2.22616e-21*ii,
     ------------------------------------------------------------------------
     5.24083e-21+5.80899e-21*ii, .330083+.265624*ii}}

o5 : List

Ways to use bertiniSample :

bertiniSample(ZZ,WitnessSet)

For the programmer

The object bertiniSample is a method function with options.