rationalPoints -- Compute all of the rational points of an affine variety

• Usage:

  l = rationalPoints I

• Inputs:
  • I, which is an ideal contained in a polynomial ring over a finite field.
• Optional inputs:
  • UseGB => a Boolean value, default value false, turns on and off a Groebner basis computation of the ideal. Default is false.
  • UseMinGens => a Boolean value, default value false, turns on and off a mingens computation of the ideal that may change the chosen generators.
  • SortGens => a Boolean value, default value false, sorts generators in order to make searching for zeroes more efficient.
  • LowMem => a Boolean value, default value false, uses an alternative algorithm that is slower but much less memory intensive.
  • Amount => a Boolean value, default value false, output changes to the number of zeroes.
  • Verbose => a Boolean value, default value false, output includes the generators of the ideal that the computation uses. These may be modified by UseGB or UseMinGens
• Outputs:
  • l, a list, a list of lists. Each internal list is an n-tuple of elements of the finite field such that the n-tuple represents a point in Affine n-space lying on the variety defined by the input ideal I.

i1 : R = ZZ/5[x_1..x_4];

i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o2 : Ideal of R

i3 : p = rationalPoints I

o3 = {{-2, 1, 1, 3}, {2, 4, 1, 3}, {-2, 1, 3, 1}, {2, 4, 3, 1}, {-2, 1, 2,
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     4}, {2, 4, 2, 4}, {-2, 1, 4, 2}, {2, 4, 4, 2}}

o3 : List