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<!DOCTYPE html> <html lang="en"> <head> <title>gamma -- the constant in the gamma trick, optional input for trackPaths</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="a package for Polynomial Homotopy Continuation" href="index.html">PHCpack</a> :: <a title="the constant in the gamma trick, optional input for trackPaths" href="_gamma.html">gamma</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="_interactive.html">next</a> | <a href="_factor__Witness__Set_lp..._cm__Verbose_eq_gt..._rp.html">previous</a> | <a href="_interactive.html">forward</a> | <a href="_factor__Witness__Set_lp..._cm__Verbose_eq_gt..._rp.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>gamma -- the constant in the gamma trick, optional input for trackPaths</h1> <div> <h2>Description</h2> <div> <p>The `gamma trick' refers to the following idea:</p> <p>If the solutions of the start system are regular, then we avoid singular solutions along the paths by multiplying the start system in the homotopy with a random complex constant <span class="tt">gamma</span>. This option allows the user to give a specific value of this gamma constant.</p> <p></p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = CC[x,y];</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : f = { x^3*y^5 + y^2 + x^2*y, x*y + x^2 - 1};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : (m,q,qsols) = mixedVolume(f,StartSystem=>true);</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : fsols = trackPaths(f,q,qsols,gamma => exp(ii*pi/3)) o4 = {{.742585+.425943*ii, .270685-1.00715*ii}, {-1.59272, .964857}, ------------------------------------------------------------------------ {.742585-.425943*ii, .270685+1.00715*ii}, {-.894935-.624334*ii, ------------------------------------------------------------------------ .143333+1.14868*ii}, {1.33076-.335184*ii, -.62414+.513163*ii}, ------------------------------------------------------------------------ {-.764107, -.544612}, {1.33076+.335184*ii, -.62414-.513163*ii}, ------------------------------------------------------------------------ {-.894935+.624334*ii, .143333-1.14868*ii}} o4 : List</code></pre> </td> </tr> </table> <div> <p></p> <p>Reference: {A.J. Sommese, J. Verschelde, and C.W. Wampler. <em>Introduction to numerical algebraic geometry.</em> In: Solving Polynomial Equations. Foundations, Algorithms and Applications, volume 14 of Algorithms and Computation in Mathematics, pages 301-337. Springer-Verlag, 2005.}</p> </div> </div> <div> <div> <h2>Functions with optional argument named <span class="tt">gamma</span>:</h2> <ul> <li><span><a title="Option to specify the value of the constant in the gamma trick" href="_track__Paths_lp..._cmgamma_eq_gt..._rp.html">trackPaths(...,gamma=>...)</a> -- Option to specify the value of the constant in the gamma trick</span></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="the constant in the gamma trick, optional input for trackPaths" href="_gamma.html">gamma</a> is <span>a <a title="the class of all symbols" href="../../Macaulay2Doc/html/___Symbol.html">symbol</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">PHCpack/PHCpackDoc.m2:1395:0</span>.</p> </div> </div> </div> </body> </html>
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