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_dim__Via__Bezout.html
<!DOCTYPE html> <html lang="en"> <head> <title>dimViaBezout -- computes the dimension of the given ideal $I$ probabilistically</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="Obtain random points in a variety" href="index.html">RandomPoints</a> :: <a title="computes the dimension of the given ideal $I$ probabilistically" href="_dim__Via__Bezout.html">dimViaBezout</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="___Extend__Field.html">next</a> | <a href="___Dimension__Function.html">previous</a> | <a href="___Extend__Field.html">forward</a> | <a href="___Dimension__Function.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>dimViaBezout -- computes the dimension of the given ideal $I$ probabilistically</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">dimViaBezout(I)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">I</span>, <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, in a polynomial ring over a field</span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">DimensionIntersectionAttempts</span><span class="tt"> => </span><span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, <span>default value null</span>, the number of linear spaces to try before moving to the next dimension</span></li> <li><span><span class="tt">MinimumFieldSize</span><span class="tt"> => </span><span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, <span>default value null</span>, if the ambient field is smaller than this value it will automatically be replaced with an extension</span></li> <li><span><span class="tt">Homogeneous</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value true</span></span></li> <li><span><a title="When changing coordinates, whether to replace variables by general degree 1 forms, binomials, etc." href="___Replacement.html">Replacement</a><span class="tt"> => </span><span class="tt">...</span>, <span>default value Full</span>, <span>When changing coordinates, whether to replace variables by general degree 1 forms, binomials, etc.</span></span></li> <li><span><span class="tt">Verbose</span><span class="tt"> => </span><span class="tt">...</span>, <span>default value false</span></span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, d = dimension of the ideal $I$</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This intersects $V(I)$ with successively higher dimensional random linear spaces until there is an intersection. For example, if $V(I)$ intersect a random line has a point, then we expect that $V(I)$ contains a hypersurface. If there was no intersection, this function tries a 2-dimensional linear space, and so on. This greatly speeds up some computations, although in other examples, the built in <span class="tt">dim</span> function is much faster.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : kk=ZZ/101;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : S=kk[y_0..y_8];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : I=ideal random(S^1,S^{-2,-2,-2,-2})+(ideal random(2,S))^2; o3 : Ideal of S</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : elapsedTime dimViaBezout(I) -- 1.52561s elapsed o4 = 4</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : elapsedTime dim I -- 3.69138s elapsed o5 = 4</code></pre> </td> </tr> </table> <div> <p>The user may set the <span class="tt">MinimumFieldSize</span> to ensure that the field being worked over is big enough. For instance, there are relatively few linear spaces over a field of characteristic 2, and this can cause incorrect results to be provided. If no size is provided, the function tries to guess a good size based on ambient ring.</p> </div> <div> <p>If the option <span class="tt">Homogeneous=>true</span> then we use homogeneous linear spaces if the ideal itself is homogeneous. Otherwise our linear spaces are not homogeneous.</p> </div> <div> <p>The user may also specify what sort of linear forms to intersect with via the <a title="When changing coordinates, whether to replace variables by general degree 1 forms, binomials, etc." href="___Replacement.html">Replacement</a> option.</p> </div> </div> <div> <h2>See also</h2> <ul> <li><span><a title="an option for specifying custom dimension functions" href="___Dimension__Function.html">DimensionFunction</a> -- an option for specifying custom dimension functions</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">dimViaBezout</span>:</h2> <ul> <li><kbd>dimViaBezout(Ideal)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="computes the dimension of the given ideal $I$ probabilistically" href="_dim__Via__Bezout.html">dimViaBezout</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">RandomPoints.m2:2002:0</span>.</p> </div> </div> </div> </body> </html>
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