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      <h1>oscQuadrics -- find the homogeneous quadrics in the homogeneous Kuramoto ideal</h1>
      <ul>
        <li>
          <dl class="element">
            <dt>Usage: </dt>
            <dd><code class="language-macaulay2">oscQuadrics(G,R)</code></dd>
            <dd><code class="language-macaulay2">oscQuadrics(G)</code></dd>
          </dl>
        </li>
        <li>Inputs:          <ul>
            <li><span><span class="tt">G</span>, <span>a <a href="../../Graphs/html/___Graph.html">graph</a></span>, an undirected, connected graph $G$ on vertices $0, \ldots, n-1$, where $n$ is the number of vertices of $G$</span></li>
            <li><span><span class="tt">R</span>, <span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, created with <a title="create a polynomial ring for a given graph or number of oscillators" href="_osc__Ring.html">oscRing</a></span></li>
          </ul>
        </li>
        <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:          <ul>
            <li><span><span class="tt">Reduced</span><span class="tt"> => </span><span>a <a title="the class of boolean values" href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value false</span>, if true, then the angles are reduced to the first angle being 0. That is, the number of oscillators will be one less than the number of vertices of $G$</span></li>
          </ul>
        </li>
        <li>Outputs:          <ul>
            <li><span><span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, in the ring R</span></li>
          </ul>
        </li>
      </ul>
      <div>
        <h2>Description</h2>
        <div>
          <p>The ideal of quadrics of the oscillator system associated to the graph $G$. This is the ideal generated by the quadrics of the system, where the system is $d\theta_i/dt = \sum_j a_{ij} \sin(\theta_j - \theta_i)$ where we consider only reduced equilibrium solutions $\theta_0 = 0$.</p>
        </div>
        <table class="examples">
          <tr>
            <td>
              <pre><code class="language-macaulay2">i1 : G = graph({0,1,2,3}, {{0,1},{1,2},{2,3},{0,3}})

o1 = Graph{0 => {1, 3}}
           1 => {0, 2}
           2 => {1, 3}
           3 => {0, 2}

o1 : Graph</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i2 : oscRing(G, CoefficientRing => CC)

o2 = CC  [x ..y ]
       53  0   3

o2 : PolynomialRing</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i3 : S = oo

o3 = S

o3 : PolynomialRing</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i4 : I = oscQuadrics(G,S)

o4 = ideal (x y  + x y  - x y  - x y , - x y  + x y  + x y  - x y , - x y  +
             1 0    3 0    0 1    0 3     1 0    0 1    2 1    1 2     2 1  
     ------------------------------------------------------------------------
     x y  + x y  - x y , - x y  - x y  + x y  + x y )
      1 2    3 2    2 3     3 0    3 2    0 3    2 3

o4 : Ideal of S</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i5 : netList I_*

+---------------------------+
o5 = |x y  + x y  - x y  - x y   |
     | 1 0    3 0    0 1    0 3  |
     +---------------------------+
     |- x y  + x y  + x y  - x y |
     |   1 0    0 1    2 1    1 2|
     +---------------------------+
     |- x y  + x y  + x y  - x y |
     |   2 1    1 2    3 2    2 3|
     +---------------------------+
     |- x y  - x y  + x y  + x y |
     |   3 0    3 2    0 3    2 3|
     +---------------------------+</code></pre>
            </td>
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      <div>
        <div class="waystouse">
          <h2>Ways to use <span class="tt">oscQuadrics</span>:</h2>
          <ul>
            <li><kbd>oscQuadrics(Graph)</kbd></li>
            <li><kbd>oscQuadrics(Graph,Ring)</kbd></li>
          </ul>
        </div>
        <div class="waystouse">
          <h2>For the programmer</h2>
          <p>The object <a title="find the homogeneous quadrics in the homogeneous Kuramoto ideal" href="_osc__Quadrics.html">oscQuadrics</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">Oscillators/Documentation.m2:409:0</span>.</p>
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