certifyDistinctSolutions -- determine whether given points are distinct approximate solutions to the system

Synopsis

This function executes the gamma test based on the value computed by computeConstants, and determine whether given points are distinct or not.

i1 : R = RR[x1,x2,y1,y2];

i2 : f = polySystem {3*y1 + 2*y2 -1, 3*x1 + 2*x2 -3.5,x1^2 + y1^2 -1, x2^2 + y2^2 - 1};

i3 : p1 = point{{.95,.32,-.30,.95}};

i4 : p2 = point{{.65,.77,.76,-.64}};

i5 : certifyDistinctSolutions(f,p1,p2)

o5 = true

However, if two solutions are too close, it concludes that inputs are not distinct.

i6 : p1 = point{{.6525,.7712,.7577,-.6366}};

i7 : p2 = point{{.653,.771,.758,-.637}};

i8 : certifyDistinctSolutions(f,p1,p2)

o8 = false

Even worse, if two solutions are close enough and both have alpha value which are bigger than $0.03$, it gives indecisive comments.

In this case, user should apply newton to the point to get more precise approximation.

i9 : p1 = point{{.95,.32,-.30,.95}};

i10 : p2 = point{{.95,.32,-.301,.95}};

i11 : certifyDistinctSolutions(f,p1,p2)

o11 = false