One Hat Cyber Team
Your IP :
216.73.216.135
Server IP :
194.44.31.54
Server :
Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64
Server Software :
Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k
PHP Version :
5.6.40
Buat File
|
Buat Folder
Eksekusi
Dir :
~
/
usr
/
share
/
doc
/
Macaulay2
/
ExampleSystems
/
html
/
View File Name :
_cohn3.html
<!DOCTYPE html> <html lang="en"> <head> <title>cohn3 -- modular equations for special algebraic number fields</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="examples of polynomial systems" href="index.html">ExampleSystems</a> :: <a title="modular equations for special algebraic number fields" href="_cohn3.html">cohn3</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="_comb3000s.html">next</a> | <a href="_chemequ.html">previous</a> | <a href="_comb3000s.html">forward</a> | <a href="_chemequ.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>cohn3 -- modular equations for special algebraic number fields</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">cohn3(kk)</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">kk</span>, <span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, the coefficient ring</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span>a <a title="the class of all lists -- {...}" href="../../Macaulay2Doc/html/___List.html">list</a></span>, of the polynomials in the system</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>This system was solved in May 2020, using <a title="solve a system of polynomial equations" href="../../NumericalAlgebraicGeometry/html/_solve__System_lp__List_rp.html">solveSystem</a> in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.</p> <p>There were 72 solutions found in 13.8985 seconds (with a Bezout bound of 1080).</p> <p>Reference: See the PoSSo test suite. Andre' Galligo and Carlo Traverso. "Practical Determination of the dimension of an algebraic variety", in E. Kaltofen and S.M. Watt, Eds "Computers and Mathematics", pages 46-52, 1989.</p> <p>H. Cohn. "An explicit modular equation in two variables and Hilbert's Twelfth problem", Math. of Comp. 38, pp. 227-236, 1982.</p> <p>H. Cohn, J. Deutch. "An explicit modular equation in two variables for Q[sqrt(3)]", Math. of Comp. 50, pp. 557-568, 1988.</p> <p>See also: http://homepages.math.uic.edu/~jan/Demo/cohn3.html</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : F = cohn3(QQ) 3 2 2 2 2 2 2 2 2 2 2 o1 = {- x y + 2x y z - x y*z - 144x y - 207x y*z + 288x*y z + 78x*y*z + ------------------------------------------------------------------------ 3 2 2 2 x*z - 3456x y - 5184x*y - 9504x*y*z - 432x*z - 248832x*y + 62208x*z - ------------------------------------------------------------------------ 3 2 2 2 2 3 2 2 3 2 2 2 2985984x, - x z*t + x z t - 6x z*t + 4x z t + 32x t - 72x z*t - ------------------------------------------------------------------------ 2 2 3 2 3 2 2 2 2 2 87x*z t - z t - 8x z - 432x z*t - 414x*z t + 2592x*z*t + 864z t - ------------------------------------------------------------------------ 2 2 2 1728x z - 20736x*z*t + 3456z t - 186624z*t - 124416x*z - 1492992z*t - ------------------------------------------------------------------------ 2 3 2 3 3 3 2 2 2 2 3 2 2985984z, x y*t - 2x*y t + y t + 8x y*t - 12x*y t + 4y t - ------------------------------------------------------------------------ 3 2 3 2 2 2 2 2 3 24x*y*t + 24y t + 20x y*t - 20x*y t - 160x*y*t + 96y t + 128x*t + ------------------------------------------------------------------------ 2 2 3 3 2 3 16x y + 96x*y*t + 2304x*t + 1152x*y + 13824x*t + 27648x, y t - y z*t ------------------------------------------------------------------------ 3 2 2 2 2 3 3 2 2 2 2 2 + 4y t - 2y z*t + 72y t + 71y*z*t + 288y t + 360y*z*t + 6z t + ------------------------------------------------------------------------ 3 3 2 2 2 3 1728y*t - 464z*t + 432y*z*t + 8z t + 6912y*t - 4320z*t + 13824t + ------------------------------------------------------------------------ 2 2 z - 13824z*t + 55296t - 13824z} o1 : List</code></pre> </td> </tr> </table> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">cohn3</span>:</h2> <ul> <li><kbd>cohn3(Ring)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="modular equations for special algebraic number fields" href="_cohn3.html">cohn3</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">ExampleSystems/cohn3.m2:55:0</span>.</p> </div> </div> </div> </body> </html>