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<div><h1>gfan</h1>
<div class="single"><h2>Description</h2>
<div><div>This method compute the Groebner fan of an ideal, which is a list of all reduced Groebner bases of an ideal. A particular Grobner basis may occur twice if it is a Groebner basis under multiple term orderings. Rather than marking the each Groebner basis to distinguish the leading terms, a second parallel list is given which contains the lead terms of each Groebner basis.</div>
<table class="examples"><tr><td><pre>i1 : R = ZZ/32003[symbol a..symbol d];</pre>
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<tr><td><pre>i2 : I = monomialCurveIdeal(R,{1,3,4})
3 2 2 2 3 2
o2 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : time (M,L) = gfan I;
LP algorithm being used: "cddgmp".
-- used 0.00248 seconds</pre>
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<tr><td><pre>i4 : M/toString/print;
{b*d^2, a*d, a*c^2, a^2*c}
{c^3, a*d, a*c^2, a^2*c}
{c^3, b*c, a*c^2, a^2*c, a^3*d}
{c^3, b*c, b^4, a*c^2, a^2*c}
{c^3, b*c, b^3, a*c^2}
{c^3, b*c, b^2*d, b^3}
{b*d^2, b^2*d, a*d, a^2*c}
{b*d^2, b^2*d, b^3, a*d}
{b*d^2, b*c, b^2*d, b^3, a*d^3}
{c^4, b*d^2, b*c, b^2*d, b^3}</pre>
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<tr><td><pre>i5 : L/toString/print;
{-c^3+b*d^2, -b*c+a*d, a*c^2-b^2*d, -b^3+a^2*c}
{c^3-b*d^2, -b*c+a*d, a*c^2-b^2*d, -b^3+a^2*c}
{c^3-b*d^2, b*c-a*d, a*c^2-b^2*d, -b^3+a^2*c, -b^4+a^3*d}
{c^3-b*d^2, b*c-a*d, b^4-a^3*d, a*c^2-b^2*d, -b^3+a^2*c}
{c^3-b*d^2, b*c-a*d, b^3-a^2*c, a*c^2-b^2*d}
{c^3-b*d^2, b*c-a*d, -a*c^2+b^2*d, b^3-a^2*c}
{-c^3+b*d^2, -a*c^2+b^2*d, -b*c+a*d, -b^3+a^2*c}
{-c^3+b*d^2, -a*c^2+b^2*d, b^3-a^2*c, -b*c+a*d}
{-c^3+b*d^2, b*c-a*d, -a*c^2+b^2*d, b^3-a^2*c, -c^4+a*d^3}
{c^4-a*d^3, -c^3+b*d^2, b*c-a*d, -a*c^2+b^2*d, b^3-a^2*c}</pre>
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<p>We can see that the leading terms of <tt>-c<sup>3</sup>+b*d<sup>2</sup>, -b*c+a*d, a*c<sup>2</sup>-b<sup>2</sup>*d, -b<sup>3</sup>+a<sup>2</sup>*c</tt> (which is the first Groebner basis listed in <tt>L</tt>) are <tt>b*d<sup>2</sup>, a*d, a*c<sup>2</sup>, a<sup>2</sup>*c</tt>.</p>
<div>If the ideal is invariant under some permutation of the variables, then gfan can compute all initial ideals up to this equivalence, which can change an intractible problem to a doable one. The cyclic group of order 4 a --> b --> c --> d --> a leaves the following ideal fixed.</div>
<table class="examples"><tr><td><pre>i6 : S = ZZ/32003[a..e];</pre>
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<tr><td><pre>i7 : I = ideal"a+b+c+d,ab+bc+cd+da,abc+bcd+cda+dab,abcd-e4"
o7 = ideal (a + b + c + d, a*b + b*c + a*d + c*d, a*b*c + a*b*d + a*c*d +
------------------------------------------------------------------------
4
b*c*d, a*b*c*d - e )
o7 : Ideal of S</pre>
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<tr><td><pre>i8 : (inL,L) = gfan I;
LP algorithm being used: "cddgmp".</pre>
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<tr><td><pre>i9 : #inL
o9 = 96</pre>
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<div>There are 96 initial ideals of this ideal. We can use the symmetry on the above:</div>
<table class="examples"><tr><td><pre>i10 : (inL1, L1) = gfan(I, Symmetries=>{(b,c,d,a,e)});
LP algorithm being used: "cddgmp".</pre>
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<tr><td><pre>i11 : #inL1
o11 = 24</pre>
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<div class="waystouse"><h2>Ways to use <tt>gfan</tt> :</h2>
<ul><li>gfan(Ideal)</li>
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