# projectiveWitnessSet -- construct a ProjectiveWitnessSet

## Synopsis

• Usage:
w = projectiveWitnessSet(E,C,S,P)
• Inputs:
• E, an ideal, in a polynomial ring over CC
• C, , in a polynomial ring over CC
• S, , complex coefficients of a linear system
• P, a list, contains witness points (of type Point)
• Outputs:

## 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 .