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<head><title>rationalPoints -- Compute all of the rational points of an affine variety</title>
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<div><h1>rationalPoints -- Compute all of the rational points of an affine variety</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>l = rationalPoints I</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, which is <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span> contained in a polynomial ring over a finite field.</span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>l</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, 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 <tt>I</tt>.</span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>UseGB => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, turns on and off a Groebner basis computation of the ideal. Default is false.</span></span></li>
<li><span><tt>UseMinGens => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, turns on and off a mingens computation of the ideal that may change the chosen generators.</span></span></li>
<li><span><tt>SortGens => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, sorts generators in order to make searching for zeroes more efficient.</span></span></li>
<li><span><tt>LowMem => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, uses an alternative algorithm that is slower but much less memory intensive.</span></span></li>
<li><span><tt>Amount => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, output changes to the number of zeroes.</span></span></li>
<li><span><tt>Verbose => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, output includes the generators of the ideal that the computation uses. These may be modified by UseGB or UseMinGens</span></span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : R = ZZ/5[x_1..x_4];</pre>
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<tr><td><pre>i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o2 : Ideal of R</pre>
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<tr><td><pre>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</pre>
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<p>This symbol is provided by the package <a href="index.html" title="Find the rational points of an affine variety defined over a finite field">RationalPoints</a>.</p>
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<div class="waystouse"><h2>Ways to use <tt>rationalPoints</tt> :</h2>
<ul><li>rationalPoints(Ideal)</li>
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