randomPoints -- a function to find random points in a variety.

Synopsis

• Usage:

  randomPoints(I)

  randomPoints(n, I)

• Inputs:
  • n, an integer, an integer denoting the number of desired points.
  • I, an ideal, inside a polynomial ring.
  • R, a ring, a polynomial ring
• Optional inputs:
  • Strategy => a symbol, default value Default, to specify which strategy to use, Default, BruteForce, LinearIntersection, GenericProjection, HybridProjectionIntersection
  • ProjectionAttempts => an integer, default value 30, see ProjectionAttempts
  • MaxCoordinatesToReplace => an integer, default value 1, see MaxCoordinatesToReplace
  • Codimension => an integer, default value null, see Codimension
  • ExtendField => a Boolean value, default value false, whether to allow points not rational over the base field
  • IntersectionAttempts => an integer, default value 20, see IntersectionAttempts PointCheckAttempts => ZZ points to search in total, see PointCheckAttempts
  • NumThreadsToUse => an integer, default value 1, number of threads to use in the BruteForce strategy, see NumThreadsToUse
  • Homogeneous => ..., default value true
  • MaxCoordinatesToTrivialize => ..., default value infinity, the number of coordinates to set to random values when doing a linear intersection
  • PointCheckAttempts => ..., default value 100, Number of times the the function will search for a point
  • Replacement => ..., default value Binomial, When changing coordinates, whether to replace variables by general degre 1 forms or binomials
  • Verbose => ..., default value false, turns out Verbose (debugging) output
• Outputs:
  • a list, a list of points in the variety with possible repetitions.

Description

i1 : R = ZZ/5[t_1..t_3];

i2 : I = ideal(t_1,t_2+t_3);

o2 : Ideal of R

i3 : randomPoints(3, I)

o3 = {{0, -2, 2}, {0, 1, -1}, {0, 2, -2}}

o3 : List

i4 : randomPoints(4, I, Strategy => Default)

o4 = {{0, -1, 1}, {0, 1, -1}, {0, -2, 2}, {0, -1, 1}}

o4 : List

i5 : randomPoints(4, I, Strategy => GenericProjection)

o5 = {{0, -1, 1}, {0, 1, -1}, {0, -1, 1}, {0, -1, 1}}

o5 : List

i6 : randomPoints(4, I, Strategy => LinearIntersection)

o6 = {{0, -2, 2}, {0, -2, 2}, {0, -1, 1}, {0, 2, -2}}

o6 : List