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# projectiveWitnessSet -- construct a ProjectiveWitnessSet

- 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 )

## Ways to use projectiveWitnessSet:

- `projectiveWitnessSet(Ideal,Matrix,Matrix,List)`

## For the programmer

The object projectiveWitnessSet is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NAGtypes/doc-NAGtypes.m2:655:0.