alphaTheoryCertification -- executes alpha-certification on a given system and list of points

• Usage:

  (D, R, CS, C) = alphaTheoryCertification(PS, L)

• Inputs:
  • PS, a polynomial system,
  • L, a list,
• Outputs:
  • D, a list, a list of certified distinct solutions
  • R, a list, a list of certified real solutions
  • CS, a list, a list of certified solutions
  • C, a list, a list of constants for certified solutions

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{{.954379,.318431,-.298633,.947949}}; p2 = point{{.95, .32, -.30, .95}}; p3 = point{{.652567, .77115, .757776, -.636663}}; p4 = point{{.65, .77, .76, -.64}};

i7 : p5 = point{{.31, .30, .72, -.60}}; -- poorly approximated solution

i8 : P = {p1, p2, p3, p4, p5}

o8 = {p1, p2, p3, p4, p5}

o8 : List

i9 : alphaTheoryCertification(f,P)

o9 = MutableHashTable{...4...}

o9 : MutableHashTable