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<!DOCTYPE html> <html lang="en"> <head> <title>msolveRUR -- compute the rational univariate representation using symbolic methods</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="Macaulay2 interface for msolve; computes real solutions and Groebner basis, etc." href="index.html">Msolve</a> :: <a title="compute the rational univariate representation using symbolic methods" href="_msolve__R__U__R.html">msolveRUR</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="_msolve__Saturate.html">next</a> | <a href="_msolve__Real__Solutions.html">previous</a> | <a href="_msolve__Saturate.html">forward</a> | <a href="_msolve__Real__Solutions.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>msolveRUR -- compute the rational univariate representation using symbolic methods</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">msolveRUR(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>, which is zero dimensional, in a polynomial ring with coefficients over <a title="the class of all rational numbers" href="../../Macaulay2Doc/html/___Q__Q.html">QQ</a></span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">Threads</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 5</span>, number of processor threads to use; </span></li> <li><span><span class="tt">Verbosity</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 0</span>, level of verbosity between 0, 1, and 2; </span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">RUR</span>, <span>a <a title="the class of all hash tables" href="../../Macaulay2Doc/html/___Hash__Table.html">hash table</a></span>, with 6 keys giving the rational univariate representation of I</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This functions uses the msolve package to compute the rational univariate representation (RUR) of a zero dimensional polynomial ideal with either integer or rational coefficients.</p> <p>The RUR gives a parametrization for all complex solutions to the input system. For a complete definition of the RUR see the paper: Rouillier, Fabrice (1999). "Solving Zero-Dimensional Systems Through the Rational Univariate Representation". Appl. Algebra Eng. Commun. Comput. 9 (9): 433–461.</p> <p>If I is a zero dimensional ideal in QQ[x_1..x_n] then the RUR is given by:</p> <p>(x_1,..,x_n)={ (-v_1(T)/w'(T), .. , -v_n(T)/w'(T)) | w(T)=0}</p> <p>The output is a hash table with 6 keys.</p> <p>The key "degree" is the number of solutions to I, counted with multiplicity.</p> <p>The key "findRootsUniPoly" gives the polynomial w(T) above.</p> <p>The key "denominator" gives the polynomial w'(T), which is the derivative of w(T) and is the denominator of each coordinate above.</p> <p>The key "numerator" gives a list {v_1(T), .. , v_n(T)} of length n above, with n the number of variables, where the polynomial v_i(T) gives the numerator of the ith coordinate.</p> <p>The key "var" gives the variable name in the univariate polynomial ring; by default this is: "T".</p> <p>The key "T" gives the linear relation between the variables of the ring of I and the single variable, which is denoted T above.</p> <p></p> </div> <div> <p>A simple example, where the input ideal is zero dimensional and radical.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[x_1..x_3] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : f = (x_1-1) o2 = x - 1 1 o2 : R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : g = (x_2-2) o3 = x - 2 2 o3 : R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : h = (x_3^2-9) 2 o4 = x - 9 3 o4 : R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : I = ideal (f,g,h) 2 o5 = ideal (x - 1, x - 2, x - 9) 1 2 3 o5 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : decompose I o6 = {ideal (x - 3, x - 2, x - 1), ideal (x + 3, x - 2, x - 1)} 3 2 1 3 2 1 o6 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : rur=msolveRUR(I) o7 = HashTable{"degree" => 2 } "denominator" => 2T 2 "findRootsUniPoly" => T - 9 2 "numerator" => {-2T, -4T, -2T } "T" => x 3 "var" => T o7 : HashTable</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : factor rur#"findRootsUniPoly" o8 = (T - 3)(T + 3) o8 : Expression of class Product</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : sols=-1*(rur#"numerator") 2 o9 = {2T, 4T, 2T } o9 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : denom= rur#"denominator" o10 = 2T o10 : QQ[T]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : (for s in sols list sub(s,T=>3))/sub(denom,T=>3) o11 = {1, 2, 3} o11 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : (for s in sols list sub(s,T=>-3))/sub(denom,T=>-3) o12 = {1, 2, -3} o12 : List</code></pre> </td> </tr> </table> <div> <p>In cases where the input ideal has dimension greater than zero an error will be returned.</p> <p></p> </div> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">msolveRUR</span>:</h2> <ul> <li><kbd>msolveRUR(Ideal)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="compute the rational univariate representation using symbolic methods" href="_msolve__R__U__R.html">msolveRUR</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">Msolve.m2:636:0</span>.</p> </div> </div> </div> </body> </html>