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<!DOCTYPE html> <html lang="en"> <head> <title>testElement -- find a test element of a ring</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="a package for calculations of singularities in positive characteristic " href="index.html">TestIdeals</a> :: <a title="find a test element of a ring" href="_test__Element.html">testElement</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="_test__Ideal.html">next</a> | <a href="___Substitution.html">previous</a> | <a href="_test__Ideal.html">forward</a> | <a href="___Substitution.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>testElement -- find a test element of a ring</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">testElement(R)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">R</span>, <span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, a quotient of a polynomial ring</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">AssumeDomain</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>, assumes the ring is a domain</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all ring elements handled by the engine" href="../../Macaulay2Doc/html/___Ring__Element.html">ring element</a></span>, a test element of <span class="tt">R</span></span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>Given $R = S/I$, where $S$ is a polynomial ring, the function <span class="tt">testElement</span> finds an element of the ambient ring $S$ whose image in $R$ is a test element of $R$. This is done by finding a minor of the jacobian of $I$ that does not lie in any minimal prime of $I$. This function considers random minors until one is found, instead of computing all minors. Thus, repeated calls will not always produce the same answer.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = ZZ/11[x,y,z]/(x^3 + y^3 + z^3);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : apply(1..5, i -> testElement(R)) 2 2 2 2 2 o2 = (2y , 2x , -3x , -4x , -y ) o2 : Sequence</code></pre> </td> </tr> </table> <div> <p>If the option <span class="tt">AssumeDomain</span> (default value <span class="tt">false</span>) is set to <span class="tt">true</span>, then <span class="tt">testElement</span> does not compute the minimal primes of $I$. This can result in a substantial speedup in some cases.</p> </div> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">testElement</span>:</h2> <ul> <li><kbd>testElement(Ring)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="find a test element of a ring" href="_test__Element.html">testElement</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">TestIdeals/testIdealsDoc.m2:73:0</span>.</p> </div> </div> </div> </body> </html>