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<div><h1>gin -- the generic initial ideal</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> gin I</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, an <a href="../../Macaulay2Doc/html/___Ideal.html" title="the class of all ideals">Ideal</a> in a polynomial<a href="../../Macaulay2Doc/html/___Ring.html" title="the class of all rings">Ring</a></span></li>
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<li><div class="single">Outputs:<ul><li><span>an <a href="../../Macaulay2Doc/html/___Ideal.html" title="the class of all ideals">Ideal</a>, the generic initial ideal of <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>AttemptCount => </tt><span><span>default value 7</span>, sets the number of random coordinate changes the routine attempts before choosing the potential<tt>gin</tt> .</span></span></li>
<li><span><tt>Modular => </tt><span><span>default value false</span>, if set to be true, computations are performed modulo a large random prime .</span></span></li>
<li><span><tt>MonomialOrder => </tt><span><span>default value null</span>, sets the<tt>Monomial Order </tt> used in the computation of <tt>gin </tt>.</span></span></li>
<li><span><tt>Verbose => </tt><span><span>default value false</span>, provides a summary of the random initial ideals generated and warns if the selected one is not strongly stable.</span></span></li>
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<div class="single"><h2>Description</h2>
<div><p>Example: a complete intersection of type (3,3) in P^3</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : I = ideal(a^3+c^2*d, b^3-a*d^2);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : gin(I)
3 2 3 5
o3 = ideal (a , a b, a*b , b )
o3 : Ideal of R</pre>
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<p>The Stanley-Reisner ideal of RP^2</p>
<table class="examples"><tr><td><pre>i4 : loadPackage "GenericInitialIdeal"
o4 = GenericInitialIdeal
o4 : Package</pre>
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<tr><td><pre>i5 : R = QQ[x0,x1,x2,x3,x4,x5]
o5 = R
o5 : PolynomialRing</pre>
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<tr><td><pre>i6 : M = matrix {{x1*x3*x4, x0*x3*x4, x1*x2*x4, x0*x2*x3, x0*x1*x2, x2*x4*x5, x0*x4*x5, x2*x3*x5, x1*x3*x5, x0*x1*x5}} --Stanley-Reisner ideal of RP^2
o6 = | x1x3x4 x0x3x4 x1x2x4 x0x2x3 x0x1x2 x2x4x5 x0x4x5 x2x3x5 x1x3x5 x0x1x5
------------------------------------------------------------------------
|
1 10
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : I=ideal flatten entries M
o7 = ideal (x1*x3*x4, x0*x3*x4, x1*x2*x4, x0*x2*x3, x0*x1*x2, x2*x4*x5,
------------------------------------------------------------------------
x0*x4*x5, x2*x3*x5, x1*x3*x5, x0*x1*x5)
o7 : Ideal of R</pre>
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<tr><td><pre>i8 : J=(ideal{x0,x1,x2})^3
3 2 2 2 2 3 2 2
o8 = ideal (x0 , x0 x1, x0 x2, x0*x1 , x0*x1*x2, x0*x2 , x1 , x1 x2, x1*x2 ,
------------------------------------------------------------------------
3
x2 )
o8 : Ideal of R</pre>
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<tr><td><pre>i9 : assert(gin(I)==J)</pre>
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<p>This symbol is provided by the package <a href="index.html" title="">GenericInitialIdeal</a>.</p>
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<div class="single"><h2>Caveat</h2>
<div>The method <tt> gin</tt> uses a probabilistic algorithm. The returned answer is correct with high probability in characteristic zero and large positive characteristic, but might be wrong in small positive characteristic. For details in this situation it is recommended to use the Verbose option.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_lexgin.html" title="the generic initial ideal with respect to lexicographical order">lexgin</a> -- the generic initial ideal with respect to lexicographical order</span></li>
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<div class="waystouse"><h2>Ways to use <tt>gin</tt> :</h2>
<ul><li><span><a href="_gin_lp__Ideal_rp.html" title="the generic initial ideal">gin(Ideal)</a> -- the generic initial ideal</span></li>
<li><span><a href="_gin_lp__Quotient__Ring_rp.html" title="the generic initial ideal">gin(QuotientRing)</a> -- the generic initial ideal</span></li>
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