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<!DOCTYPE html> <html lang="en"> <head> <title>msolveGB -- compute generators of a Groebner basis in GRevLex order</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 generators of a Groebner basis in GRevLex order" href="_msolve__G__B.html">msolveGB</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__Lead__Monomials.html">next</a> | <a href="_msolve__Eliminate.html">previous</a> | <a href="_msolve__Lead__Monomials.html">forward</a> | <a href="_msolve__Eliminate.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>msolveGB -- compute generators of a Groebner basis in GRevLex order</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">msolveGB(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>, in a polynomial ring with <a title="graded reverse lexicographical monomial ordering" href="../../Macaulay2Doc/html/___G__Rev__Lex.html">GRevLex</a> order and coefficients over <a title="the class of all rational numbers" href="../../Macaulay2Doc/html/___Q__Q.html">QQ</a> or <a href="../../Macaulay2Doc/html/_finite_spfields.html">ZZ/p</a> in characteristic less than $2^{31}$</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">GB</span>, <span>a <a title="the class of all matrices" href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose columns form a Groebner basis for the input ideal I, in the GRevLex order</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This functions uses the F4 implementation in the msolve package to compute a Groebner basis, in GRevLex order, of a polynomial ideal with either rational coefficients or finite field coefficients with characteristic less than $2^{31}$. If the input ideal is a polynomial ring with monomial order other than GRevLex a GRevLex basis is returned (and no warning is given). The input ideal may also be given in a ring with integer coefficients, in this case a Groebner basis for the given ideal over the rationals will be computed, denominators will be cleared, and the output will be a Groebner basis over the rationals in GRevLex order with integer coefficients.</p> </div> <div> <p>First an example over a finite field</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R=ZZ/1073741827[z_1..z_3] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I=ideal(7*z_1*z_2+5*z_2*z_3+z_3^2+z_1+5*z_3+10,8*z_1^2+13*z_1*z_3+10*z_3^2+z_2+z_1) 2 2 2 o2 = ideal (7z z + 5z z + z + z + 5z + 10, 8z + 13z z + 10z + z + 1 2 2 3 3 1 3 1 1 3 3 1 ------------------------------------------------------------------------ z ) 2 o2 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : gB=msolveGB I o3 = | z_1z_2-460175068z_2z_3-306783379z_3^2-306783379z_1-460175068z_3+ ------------------------------------------------------------------------ 153391691 z_1^2+134217730z_1z_3+268435458z_3^2-402653185z_1-402653185z_2 ------------------------------------------------------------------------ z_1z_3^2+19173957z_2z_3^2-479349029z_3^3-402653186z_2^2+460175073z_1z_3+ ------------------------------------------------------------------------ 134217729z_2z_3+153391693z_3^2+10z_1+402653185z_2-364305253z_3+268435458 ------------------------------------------------------------------------ z_2^2z_3^2+59398484z_2z_3^3+347252676z_3^4+516309900z_2^3+91382283z_2^2z ------------------------------------------------------------------------ _3+27414685z_2z_3^2+223886594z_3^3-466049644z_2^2-68536713z_2z_3+ ------------------------------------------------------------------------ 310699764z_3^2-310699763z_2+411220277z_3-68536709 | 1 4 o3 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : lT=monomialIdeal leadTerm gB 2 2 2 2 o4 = monomialIdeal (z , z z , z z , z z ) 1 1 2 1 3 2 3 o4 : MonomialIdeal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : degree lT o5 = 4</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : dim lT o6 = 1</code></pre> </td> </tr> </table> <div> <p>Now the same example over the rationals.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i7 : R=QQ[z_1..z_3] o7 = R o7 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : I=ideal(7*z_1*z_2+5*z_2*z_3+z_3^2+z_1+5*z_3+10,8*z_1^2+13*z_1*z_3+10*z_3^2+z_2+z_1) 2 2 2 o8 = ideal (7z z + 5z z + z + z + 5z + 10, 8z + 13z z + 10z + z + 1 2 2 3 3 1 3 1 1 3 3 1 ------------------------------------------------------------------------ z ) 2 o8 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : gB=msolveGB I o9 = | 7z_1z_2+5z_2z_3+z_3^2+z_1+5z_3+10 8z_1^2+13z_1z_3+10z_3^2+z_1+z_2 ------------------------------------------------------------------------ 56z_1z_3^2-235z_2z_3^2+51z_3^3-49z_2^2+240z_1z_3+35z_2z_3+192z_3^2+560z_ ------------------------------------------------------------------------ 1-7z_2+545z_3+70 235z_2^2z_3^2-11z_2z_3^3+8z_3^4+49z_2^3-35z_2^2z_3+13z_ ------------------------------------------------------------------------ 2z_3^2+67z_3^3+14z_2^2-150z_2z_3+304z_3^2-69z_2+665z_3+790 | 1 4 o9 : Matrix R <-- R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : (ideal gB)== ideal(groebnerBasis I) o10 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : lT=monomialIdeal leadTerm gB 2 2 2 2 o11 = monomialIdeal (z , z z , z z , z z ) 1 1 2 1 3 2 3 o11 : MonomialIdeal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i12 : degree lT o12 = 4</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i13 : dim lT o13 = 1</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">msolveGB</span>:</h2> <ul> <li><kbd>msolveGB(Ideal)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="compute generators of a Groebner basis in GRevLex order" href="_msolve__G__B.html">msolveGB</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>