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<!DOCTYPE html> <html lang="en"> <head> <title>hyperdeterminant -- computes the hyperdeterminant of a boundary format tensor</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="multilinear algebra for the construction of tensor complexes" href="index.html">TensorComplexes</a> :: <a title="computes the hyperdeterminant of a boundary format tensor" href="_hyperdeterminant.html">hyperdeterminant</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="_hyperdeterminant__Matrix.html">next</a> | <a href="_from__Ordinal.html">previous</a> | <a href="_hyperdeterminant__Matrix.html">forward</a> | <a href="_from__Ordinal.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>hyperdeterminant -- computes the hyperdeterminant of a boundary format tensor</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">hyperdeterminant f</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">f</span>, <span>a <a title="the class of maps between LabeledModules" href="___Labeled__Module__Map.html">map of labeled modules</a></span>, </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>, </span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This constructs the hyperdeterminant of a tensor of <em>boundary format</em>, where we say that a $a\times b_1\times \dots \times b_n$ has boundary format if $$ a-\sum_{i=1}^n (b_i-1)=1. $$ We construct the hyperdeterminant as the determinant of a certain square matrix derived from $f$. The <span class="tt">hyperdeterminant</span> function outputs the hyperdeterminant itself, whereas the <a title="computes a matrix whose determinant equals the hyperdeterminant of a boundary format tensor" href="_hyperdeterminant__Matrix.html">hyperdeterminantMatrix</a> function outputs the matrix used to compute the hyperdeterminant. (For background on computing hyperdeterminants, see Section 14.3 of the book ``Discriminants, resultants, and multidimensional determinants '' by Gelfand-Kapranov-Zelevinsky.)</p> <p>The following constructs the generic hyperdeterminant of format $3\times 2\times 2$, which is a polynomial of degree 6 consisting of 66 monomials.</p> <p></p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : f=flattenedGenericTensor({3,2,2},QQ); 4 3 o1 : Matrix (QQ[x ..x ]) <--- (QQ[x ..x ]) 0,0,0 2,1,1 0,0,0 2,1,1</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : S=ring f;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : h=hyperdeterminant f;</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : degree h o4 = {6} o4 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : #terms h o5 = 66</code></pre> </td> </tr> </table> </div> <div> <h2>Caveat</h2> <div> <p>There is bug involving the graded structure of the output. Namely, the code assumes that all entries of <span class="tt">f</span> have degree 1, and gives the wrong graded structure if this is not the case. If <span class="tt">ring f</span> is not graded, then the code gives an error.</p> <p></p> </div> </div> <div> <h2>See also</h2> <ul> <li><span><a title="computes a matrix whose determinant equals the hyperdeterminant of a boundary format tensor" href="_hyperdeterminant__Matrix.html">hyperdeterminantMatrix</a> -- computes a matrix whose determinant equals the hyperdeterminant of a boundary format tensor</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">hyperdeterminant</span>:</h2> <ul> <li><kbd>hyperdeterminant(LabeledModuleMap)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="computes the hyperdeterminant of a boundary format tensor" href="_hyperdeterminant.html">hyperdeterminant</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">TensorComplexes.m2:955:0</span>.</p> </div> </div> </div> </body> </html>