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<!DOCTYPE html> <html lang="en"> <head> <title>gbs -- get the list of Groebner bases corresponding to the height-one primes in the support of a divisor</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="divisors" href="index.html">Divisor</a> :: <a title="get the list of Groebner bases corresponding to the height-one primes in the support of a divisor" href="_gbs.html">gbs</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="_get__Linear__Diophantine__Solution.html">next</a> | <a href="_find__Element__Of__Degree.html">previous</a> | <a href="_get__Linear__Diophantine__Solution.html">forward</a> | <a href="_find__Element__Of__Degree.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>gbs -- get the list of Groebner bases corresponding to the height-one primes in the support of a divisor</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">gbs( D1 )</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">D1</span>, <span>an instance of the type <a title="the Types of divisors" href="___Basic__Divisor.html">BasicDivisor</a></span>, </span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This function returns the list of Groebner bases associated to the height-one prime ideals corresponding to the components of a <span class="tt">BasicDivisor</span> (or a <span class="tt">WeilDivisor</span>, <span class="tt">QWeilDivisor</span> or <span class="tt">RWeilDivisor</span>). Note that this list of Groebner bases is made when the divisor is constructed.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = ZZ/7[x,y,u,v]/ideal(x*y-u*v);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : D = divisor(x) o2 = Div(v, x) + Div(u, x) o2 : WeilDivisor on R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : gbs(D) o3 = {{v, x}, {u, x}} o3 : List</code></pre> </td> </tr> </table> <div> <p>Note, the Grobner basis can be different from a minimal set of generators the user provides.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : R = ZZ/2[x,y,z]/ideal(z^2+x*y*z+x^2*y+x*y^2);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : J = ideal(x+y, x^2+z); o5 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : D = divisor({2}, {J}) o6 = 2*Div(x+y, x^2+z) o6 : WeilDivisor on R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : gbs(D) 2 o7 = {{x + y, y + z}} o7 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : primes(D) 2 o8 = {ideal (x + y, x + z)} o8 : List</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="get the list of coefficients of a divisor" href="_coefficients_lp__Basic__Divisor_rp.html">coefficients(BasicDivisor)</a> -- get the list of coefficients of a divisor</span></li> <li><span><a title="get the list of height-one primes in the support of a divisor" href="_primes.html">primes</a> -- get the list of height-one primes in the support of a divisor</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">gbs</span>:</h2> <ul> <li><kbd>gbs(BasicDivisor)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="get the list of Groebner bases corresponding to the height-one primes in the support of a divisor" href="_gbs.html">gbs</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">Divisor.m2:2304:0</span>.</p> </div> </div> </div> </body> </html>