projectiveWitnessSet -- construct a ProjectiveWitnessSet

Synopsis

• Usage:

  w = projectiveWitnessSet(E,C,S,P)

• Inputs:
  • E, an ideal, in a polynomial ring over CC
  • C, a matrix, in a polynomial ring over CC
  • S, a matrix, complex coefficients of a linear system
  • P, a list, contains witness points (of type AbstractPoint)
• Outputs:
  • w, an instance of the type ProjectiveWitnessSet

Description

Used to construct a witness set for a component of the variety V(E). An affine chart is specified by the matrix of the coefficients of the (normalized) linear equation defining the chart: e.g., ax+by+cz=1 is encoded as [a,b,c].

It is expected that the, V(E) and the plane V(S) defined by S are of complementary dimensions and that P is contained in the intersection of V(E+C) and V(S).

i1 : R = CC[x,y,z]	

o1 = R

o1 : PolynomialRing

i2 : w = projectiveWitnessSet( ideal(x^2+y^2+2*z^2), matrix{{0,0,1}}, matrix{{1,-1,0}}, {point {{0.999999*ii,0.999999*ii,1.}}, point {{ -1.000001*ii,-1.000001*ii,1.}}} )

o2 = w

o2 : WitnessSet

i3 : peek w

o3 = WitnessSet{AffineChart => | 0 0 1 |               }
                cache => CacheTable{...1...}
                Points => {{.999999*ii, .999999*ii, 1}}
                          {{-ii, -ii, 1}              }
                Slice => | 1 -1 0 |
                                    2    2     2
                Equations => ideal(x  + y  + 2z )