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NumericalImplicitization :: PseudoWitnessSet

PseudoWitnessSet -- the class of all PseudoWitnessSets

Description

This is a type of hashtable storing the output of a pseudo-witness set computation using monodromy, with the following keys:

  • isCompletePseudoWitnessSet: whether or not the pseudo-witness set has passed the trace test, according to the trace test threshold
  • imageDegree: the number of image points found by monodromy
  • map: the map F, of which the image is under consideration
  • sourceEquations: the defining ideal I of the source variety
  • sourceSlice: additional equations to form a zero-dimensional system (only needed if the map is not finite-to-one)
  • imageSlice: a general complementary-dimensional linear space to F(V(I))
  • witnessPointPairs: a VerticalList of 2-point sequences (p, F(p)), where p lies on V(I) and F(p) lies on imageSlice
  • traceTest: the result of the trace test applied to witnessPointPairs

The following example demonstrates the output for the 3-uple embedding of P1 into P3, whose image is the twisted cubic:

i1 : R = CC[s,t];
i2 : W = numericalImageDegree(basis(3,R), ideal 0_R, Verbose => false);
-- warning: experimental computation over inexact field begun
--          results not reliable (one warning given per session)
i3 : peek W

o3 = PseudoWitnessSet{imageDegree => 3                                                                                                                                      }
                      imageSlice => | (.025888+.714827i)y_1+(.461944+.775187i)y_2+(.074184+.808694i)y_3+(.127435+.254482i)y_4+1.84897+1.0891i |
                                    | (.89189+.231053i)y_1+(.909047+.314897i)y_2+(.362835+.706096i)y_3+(.741046+.108386i)y_4+2.04097-.925371i |
                      isCompletePseudoWitnessSet => true
                      sourceEquations => ideal 0
                      sourceSlice => 0
                      traceTest => 2.42377e-12
                      witnessPointPairs => {({.216572+.858035*ii, 1.25142+.497798*ii}, {-.468178-.510971*ii, -1.04763+.121949*ii, -.783536+1.40092*ii, 1.02945+2.21536*ii})}
                                           {({.892712+.673395*ii, .29398+.632944*ii}, {-.502996+1.3046*ii, -.660011+.57085*ii, -.531085+.120643*ii, -.327914-.0894633*ii}) }
                                           {({.023866-1.48479*ii, .280787+1.06704*ii}, {-.157831+3.27084*ii, -.543241-2.37168*ii, .864424+1.58777*ii, -.936945-.962513*ii})}
                               3   2      2   3
                      map => {s , s t, s*t , t }

[1] J. D. Hauenstein and A. J. Sommese, Witness sets of projections. Appl. Math. Comput. 217(7) (2010), 3349-3354.

See also

Functions and methods returning a pseudo-witness set :

Methods that use a pseudo-witness set :

For the programmer

The object PseudoWitnessSet is a type, with ancestor classes HashTable < Thing.