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<!DOCTYPE html> <html lang="en"> <head> <title>NumericalVariety -- a numerical variety</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="Common types used in Numerical Algebraic Geometry" href="index.html">NAGtypes</a> :: <a title="a numerical variety" href="___Numerical__Variety.html">NumericalVariety</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="_numerical__Variety_lp__List_rp.html">next</a> | <a href="_numerical__Affine__Space.html">previous</a> | <a href="_numerical__Variety_lp__List_rp.html">forward</a> | <a href="_numerical__Affine__Space.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>NumericalVariety -- a numerical variety</h1> <div> <h2>Description</h2> <p>This type stores a collection of witness sets representing a complex affine variety. </p> Note that <ul> <li>The ambient space is expected to be the same, i.e., <a title="a witness set" href="___Witness__Set.html">dimension</a> (or <a title="a witness set" href="___Witness__Set.html">ProjectionDimension</a>) of <a title="a witness set" href="___Witness__Set.html">witness sets</a> should be the same.</li> <li>The constructor <a title="construct a numerical variety" href="_numerical__Variety_lp__List_rp.html">numericalVariety(List)</a> does not check the sensibility of the input; run <a title="a numerical variety" href="___Numerical__Variety.html">check(NumericalVariety)</a> to verify the validity of a numerical variety.</li> </ul> Basic service routines: <ul> <li>dim -- the dimension</li> <li>codim -- the codimension</li> <li>deg -- the degree</li> <li><a title="list components of a numerical variety" href="_components_lp__Numerical__Variety_rp.html">components(NumericalVariety)</a></li> </ul> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = CC[x,y]; I = ideal((x^2+y^2+2)*x,(x^2+y^2+2)*y*(y-1)); o2 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : w1 := witnessSet(I , ideal(x-y), {point {{0.999*ii,0.999*ii}}, point {{-1.001*ii,-1.001*ii}}} ) o3 = [dim=1,deg=2]-*may be reducible*- o3 : WitnessSet</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : w0 := witnessSet(I, ideal R, {point {{0.,0.}}}) o4 = [dim=0,deg=1]-*may be reducible*- o4 : WitnessSet</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : w0' := witnessSet(I, ideal R, {point {{0.,1.}}}) o5 = [dim=0,deg=1]-*may be reducible*- o5 : WitnessSet</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : V := numericalVariety {w0,w1,w0'} o6 = a "numerical variety" with components in dim 0: [dim=0,deg=1]-*may be reducible*- [dim=0,deg=1]-*may be reducible*- dim 1: [dim=1,deg=2]-*may be reducible*- o6 : NumericalVariety</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : dim V o7 = 1</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : degree V o8 = 2</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="a witness set" href="___Witness__Set.html">WitnessSet</a> -- a witness set</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Functions and methods returning a numerical variety:</h2> <ul> <li><span><kbd>numericalVariety</kbd> -- see <span><a title="construct a numerical variety" href="_numerical__Variety_lp__List_rp.html">numericalVariety(List)</a> -- construct a numerical variety</span></span></li> </ul> <h2>Methods that use a numerical variety:</h2> <ul> <li><span><a title="list components of a numerical variety" href="_components_lp__Numerical__Variety_rp.html">components(NumericalVariety)</a> -- list components of a numerical variety</span></li> <li><span><kbd>components(NumericalVariety,ZZ)</kbd> -- see <span><a title="list components of a numerical variety" href="_components_lp__Numerical__Variety_rp.html">components(NumericalVariety)</a> -- list components of a numerical variety</span></span></li> <li><span><kbd>components(NumericalVariety,ZZ,InfiniteNumber)</kbd> -- see <span><a title="list components of a numerical variety" href="_components_lp__Numerical__Variety_rp.html">components(NumericalVariety)</a> -- list components of a numerical variety</span></span></li> <li><span><kbd>components(NumericalVariety,ZZ,ZZ)</kbd> -- see <span><a title="list components of a numerical variety" href="_components_lp__Numerical__Variety_rp.html">components(NumericalVariety)</a> -- list components of a numerical variety</span></span></li> <li><kbd>check(NumericalVariety)</kbd></li> <li><kbd>degree(NumericalVariety)</kbd></li> <li><kbd>dim(NumericalVariety)</kbd></li> <li><kbd>net(NumericalVariety)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="a numerical variety" href="___Numerical__Variety.html">NumericalVariety</a> is <span>a <a title="the class of all mutable types" href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a title="the class of all mutable hash tables" href="../../Macaulay2Doc/html/___Mutable__Hash__Table.html">MutableHashTable</a> < <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">HashTable</a> < <a title="the class of all things" href="../../Macaulay2Doc/html/___Thing.html">Thing</a>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">NAGtypes/doc-NAGtypes.m2:720:0</span>.</p> </div> </div> </div> </body> </html>