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<head><title>gkz -- GKZ A-hypergeometric ideal</title>
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<div><h1>gkz -- GKZ A-hypergeometric 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>gkz(A,b)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>A</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span></span></li>
<li><span><tt>b</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span></span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, which represents the Gel'fand-Kapranov-Zelevinsky hypergeometric system associated to the matrix A and the parameter vector b</span></li>
</ul>
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</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="_gkz_lp..._cm_sp__Vars_sp_eq_gt_sp..._rp.html">Vars => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>The GKZ hypergeometric system of PDE's associated to a (d x n) integer matrix A consists of the toric ideal I_A in the polynomial subring C[d_1,...,d_n] and Euler relations given by the entries of the vector (A theta - b), where theta is the vector (theta_1,...,theta_n)^t, and theta_i = x_i d_i. See the book 'Groebner deformations of hypergeometric differential equations' by Saito-Sturmfels-Takayama (1999) for more details.<table class="examples"><tr><td><pre>i1 : A = matrix{{1,1,1},{0,1,2}}
o1 = | 1 1 1 |
| 0 1 2 |
2 3
o1 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i2 : b = {3,4}
o2 = {3, 4}
o2 : List</pre>
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<tr><td><pre>i3 : I = gkz (A,b)
2
o3 = ideal (D - D D , x D + x D + x D - 3, x D + 2x D - 4)
2 1 3 1 1 2 2 3 3 2 2 3 3
o3 : Ideal of QQ[x , x , x , D , D , D ]
1 2 3 1 2 3</pre>
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<div class="single"><h2>Caveat</h2>
<div>gkz always returns a different ring and will use variables x_1,...,x_n, D_1,...D_n.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Appell__F1_lp__List_rp.html" title="Appell F1 system of PDE's">AppellF1</a> -- Appell F1 system of PDE's</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>gkz</tt> :</h2>
<ul><li>gkz(Matrix,List)</li>
</ul>
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