One Hat Cyber Team

Edit File: _gkz.html

<!DOCTYPE html>
<html lang="en">
  <head>
    <title>gkz -- create the A-hypergeometric system of Gelfand, Kapranov and Zelevinsky (GKZ)</title>
    <meta content="text/html; charset=utf-8" http-equiv="Content-Type">
    <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css">

o1 = | 1 1 1 |
     | 0 1 2 |

2       3
o1 : Matrix ZZ  &lt;-- ZZ</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i2 : b = {3,4}

o2 = {3, 4}

o2 : List</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i3 : I = gkz (A,b)

2
o3 = ideal (x D  + x D  + x D  - 3, x D  + 2x D  - 4, - D  + D D )
             1 1    2 2    3 3       2 2     3 3         2    1 3

o3 : Ideal of QQ[x ..x , D ..D ]
                  1   3   1   3</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i4 : describe ring I

o4 = QQ[x ..x , D ..D , Degrees => {6:1}, Heft => {1}, WeylAlgebra => {{x , D }, {x , D }, {x , D }}]
         1   3   1   3                                                   1   1     2   2     3   3</code></pre>
            </td>
          </tr>
        </table>
        <div>
          <p>The ambient Weyl algebra can be determined as an input.</p>
        </div>
        <table class="examples">
          <tr>
            <td>
              <pre><code class="language-macaulay2">i5 : D = makeWA(QQ[x_1..x_3])

o5 = D

o5 : PolynomialRing, 3 differential variable(s)</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i6 : gkz(A,b,D)

2
o6 = ideal (x dx  + x dx  + x dx  - 3, x dx  + 2x dx  - 4, - dx  + dx dx )
             1  1    2  2    3  3       2  2     3  3          2     1  3

o6 : Ideal of D</code></pre>
            </td>
          </tr>
        </table>
        <div>
          <p>One may separately produce the toric ideal and the Euler operators.</p>
        </div>
        <table class="examples">
          <tr>
            <td>
              <pre><code class="language-macaulay2">i7 : toricIdealPartials(A,D)

2
o7 = ideal(- dx  + dx dx )
               2     1  3

o7 : Ideal of QQ[dx ..dx ]
                   1    3</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i8 : eulerOperators(A,b,D)

o8 = {x dx  + x dx  + x dx  - 3, x dx  + 2x dx  - 4}
       1  1    2  2    3  3       2  2     3  3

o8 : List</code></pre>
            </td>
          </tr>
        </table>
      </div>
      <div>
        <h2>Caveat</h2>
        <div>
          <p>gkz(A,b) always returns a different ring and will use variables x_1,...,x_n, D_1,...D_n.</p>
        </div>
      </div>
      <div>
        <h2>See also</h2>
        <ul>
          <li><span><a title="Appell F1 system of PDE's" href="___Appell__F1.html">AppellF1</a> -- Appell F1 system of PDE's</span></li>
        </ul>
      </div>
      <div>
        <div class="waystouse">
          <h2>Ways to use <span class="tt">gkz</span>:</h2>
          <ul>
            <li><kbd>gkz(Matrix,List)</kbd></li>
            <li><kbd>gkz(Matrix,List,PolynomialRing)</kbd></li>
          </ul>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="create the A-hypergeometric system of Gelfand, Kapranov and Zelevinsky (GKZ)" href="_gkz.html">gkz</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">HolonomicSystems/DOC/Dsystems.m2:77:0</span>.</p>
        </div>
      </div>
    </div>
  </body>

</html>