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# 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.

## Description

    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,
         ------------------------------------------------------------------------
         4}, {2, 4, 2, 4}, {-2, 1, 4, 2}, {2, 4, 4, 2}}

    o3 : List

This symbol is provided by the package RationalPoints.

## Ways to use rationalPoints:

- `rationalPoints(Ideal)`

## For the programmer

The object rationalPoints is a method function with options.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/RationalPoints.m2:387:0.