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.
  • ExtendField => a Boolean value, default value false, whether to allow points not rational over the base field
  • Homogeneous => a Boolean value, default value true, whether to include the origin as a valid point to output 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
  • DimensionFunction => a function, default value dim, specify a custom dimension function, such as the default dimViaBezout or the Macaulay2 function dim
  • DecompositionStrategy => a symbol, default value null, see DecompositionStrategy
  • PointCheckAttempts => ..., default value 0, 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, binomials, etc.
  • 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, -1, 1}, {0, -1, 1}, {0, -1, 1}}

o3 : List

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

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

o4 : List

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

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

o5 : List

i6 : S=ZZ/103[y_0..y_30];

i7 : I=minors(2,random(S^3,S^{3:-1}));

o7 : Ideal of S

i8 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>MultiplicationTable)
 -- 5.04581 seconds elapsed

o8 = {{-4, -35, -7, 0, 0, 1, 5, -13, 0, -47, 0, 41, 0, -51, -46, 35, 0, 0,
     ------------------------------------------------------------------------
     -47, 14, -30, 42, 30, 4, -41, 24, 0, 0, 15, 20, 1}}

o8 : List

i9 : elapsedTime randomPoints(I, Strategy=>LinearIntersection, DecompositionStrategy=>Decompose)
 -- 2.55275 seconds elapsed

o9 = {{11, 9, -9, -15, -7, 27, 19, -36, 48, 26, -4, 3, 29, -8, 7, -32, 16,
     ------------------------------------------------------------------------
     11, 7, 7, 25, -14, -39, 17, -16, 4, -50, -12, 21, -50, 51}}

o9 : List