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___Witness__Set.html
<!DOCTYPE html> <html lang="en"> <head> <title>WitnessSet -- a witness set</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 witness set" href="___Witness__Set.html">WitnessSet</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="_witness__Set.html">next</a> | <a href="_to__Affine__Chart_lp__Z__Z_cm__List_rp.html">previous</a> | <a href="_witness__Set.html">forward</a> | <a href="_to__Affine__Chart_lp__Z__Z_cm__List_rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>WitnessSet -- a witness set</h1> <div> <h2>Description</h2> This type stores a witness set of an equidimensional solution component. The following methods can be used to access a <a title="a witness set" href="___Witness__Set.html">WitnessSet</a>: <ul> <li>ideal -- get the defining ideal of the algebraic superset</li> <li>equations -- get the list of defining polynomials of the algebraic superset</li> <li>slice -- get linear functions defining the slicing plane</li> <li>points -- get the list of witness points (which are zeroes of all above)</li> </ul> Also one may determine <ul> <li>dim -- the dimension</li> <li>codim -- the codimension</li> <li>deg -- the degree (the number of witness points)</li> <li>ring -- the ring of the defining polynomials</li> </ul> Only dimension and degree are displayed (by <a title="format for printing, as a net" href="../../Macaulay2Doc/html/_net.html">net</a>); to see the data stored in a witness set use <a title="examine contents of an object" href="../../Macaulay2Doc/html/_peek.html">peek</a>. <h2>For developers:</h2> Required keys in a <a title="a witness set" href="___Witness__Set.html">WitnessSet</a> are <ul> <li><span class="tt">Equations</span> -- <span>an <a title="the class of all ideals" href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></li> <li><span class="tt">Slice</span> -- <span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span> or <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span></li> <li><span class="tt">Points</span>-- a list of <a title="a type used to store a point in complex space" href="___Abstract__Point.html">points</a></li> <li><span class="tt">cache.IsIrreducible</span> -- takes values <a title="the unique member of the empty class" href="../../Macaulay2Doc/html/_null.html">null</a>(not determined), <a href="../../Macaulay2Doc/html/_true.html">true</a>, or <a href="../../Macaulay2Doc/html/_false.html">false</a></li> </ul> Optional keys: <ul> <li><span class="tt">ProjectionDimension</span> -- <span>an <a title="the class of all integers" href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, the witness set describes a lifted variety (its projection on the first <span class="tt">ProjectionDimension</span> coordinates is the variety the witness set represents)</li> </ul> </div> <div> <h2>See also</h2> <ul> <li><span><a title="construct a WitnessSet" href="_witness__Set.html">witnessSet</a> -- construct a WitnessSet</span></li> <li><span><a title="a projective witness set" href="___Projective__Witness__Set.html">ProjectiveWitnessSet</a> -- a projective witness set</span></li> <li><span><a title="a numerical variety" href="___Numerical__Variety.html">NumericalVariety</a> -- a numerical variety</span></li> <li><span><a title="declares a component represented by the witness set irreducible" href="_declare__Irreducible.html">declareIrreducible</a> -- declares a component represented by the witness set irreducible</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Types of witness set:</h2> <ul> <li><span><a title="a projective witness set" href="___Projective__Witness__Set.html">ProjectiveWitnessSet</a> -- a projective witness set</span></li> </ul> <h2>Functions and methods returning a witness set:</h2> <ul> <li><span><a title="construct a WitnessSet" href="_witness__Set.html">witnessSet</a> -- construct a WitnessSet</span></li> </ul> <h2>Methods that use a witness set:</h2> <ul> <li><span><kbd>declareIrreducible(WitnessSet)</kbd> -- see <span><a title="declares a component represented by the witness set irreducible" href="_declare__Irreducible.html">declareIrreducible</a> -- declares a component represented by the witness set irreducible</span></span></li> <li><span><kbd>generalEquations(WitnessSet)</kbd> -- see <span><a title="random linear combinations of equations/generators" href="_general__Equations.html">generalEquations</a> -- random linear combinations of equations/generators</span></span></li> <li><kbd>codim(WitnessSet)</kbd></li> <li><kbd>degree(WitnessSet)</kbd></li> <li><kbd>dim(WitnessSet)</kbd></li> <li><kbd>equations(WitnessSet)</kbd></li> <li><kbd>ideal(WitnessSet)</kbd></li> <li><kbd>net(WitnessSet)</kbd></li> <li><kbd>points(WitnessSet)</kbd></li> <li><kbd>ring(WitnessSet)</kbd></li> <li><kbd>slice(WitnessSet)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="a witness set" href="___Witness__Set.html">WitnessSet</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="(under construction!) new types and methods needed to generalize WitnessSet" href="___W__Set.html">WSet</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.m2:590:0</span>.</p> </div> </div> </div> </body> </html>
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