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
/
Points
/
html
/
View File Name :
index.html
<!DOCTYPE html> <html lang="en"> <head> <title>Points -- A package for making and studying points in affine and projective spaces</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 making and studying points in affine and projective spaces" href="index.html">Points</a> :: <a title="A package for making and studying points in affine and projective spaces" href="index.html">Points</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="_affine__Fat__Points.html">next</a> | previous | <a href="_affine__Fat__Points.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Points -- A package for making and studying points in affine and projective spaces</h1> <div> <h2>Description</h2> <div> <p>The package has routines for points in affine and projective spaces. The affine code, some of which uses the Buchberger-Moeller algorithm to more quickly compute the ideals of points in affine space, was written by Stillman, Smith and Stromme. The projective code was written separately by Eisenbud and Popescu.</p> <p>The purpose of the projective code was to find as many counterexamples as possible to the minimal resolution conjecture; it was of use in the research for the paper "Exterior algebra methods for the minimal resolution conjecture", by David Eisenbud, Sorin Popescu, Frank-Olaf Schreyer and Charles Walter (Duke Mathematical Journal. 112 (2002), no.2, 379-395.) The first few of these counterexamples are: (6,11), (7,12), (8,13), (10,16), where the first integer denotes the ambient dimension and the second the number of points. Examples are known in every projective space of dimension >=6 except for P^9.</p> <p>In version 3.0, F. Galetto and J.W. Skelton added code to compute ideals of fat points and projective points using the Buchberger-Moeller algorithm.</p> </div> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li><a href="https://macaulay2.com/">Mike Stillman</a><span> <<a href="mailto:mike%40math.cornell.edu">mike@math.cornell.edu</a>></span></li> <li>Stein A. Strømme<span> <<a href="mailto:stromme%40math.uib.no">stromme@math.uib.no</a>></span></li> <li>David Eisenbud<span> <<a href="mailto:de%40msri.org">de@msri.org</a>></span></li> <li><a href="http://math.galetto.org">Federico Galetto</a><span> <<a href="mailto:galetto.federico%40gmail.com">galetto.federico@gmail.com</a>></span></li> <li>Joseph W. Skelton<span> <<a href="mailto:jskelton%40tulane.edu">jskelton@tulane.edu</a>></span></li> </ul> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>3.0</b> of Points, released <b>29 June 2008, revised by DE June 2016, revised by FG and JWS June 2018</b>.</p> </div> <div> <h2>Citation</h2> <p>If you have used this package in your research, please cite it as follows:</p> <table class="examples"> <tr> <td> <pre><code class="language-bib">@misc{PointsSource, title = {{Points: sets of points. Version~3.0}}, author = {Mike Stillman and Stein A. Str{\o}mme and David Eisenbud and Federico Galetto and Joseph W. Skelton}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Functions and commands <ul> <li><span><a title="produces the ideal and initial ideal from the coordinates of a finite set of fat points" href="_affine__Fat__Points.html">affineFatPoints</a> -- produces the ideal and initial ideal from the coordinates of a finite set of fat points</span></li> <li><span><a title="computes ideal of fat points by intersecting powers of maximal ideals" href="_affine__Fat__Points__By__Intersection.html">affineFatPointsByIntersection</a> -- computes ideal of fat points by intersecting powers of maximal ideals</span></li> <li><span><a title="evaluation on points" href="_affine__Make__Ring__Maps.html">affineMakeRingMaps</a> -- evaluation on points</span></li> <li><span><a title="produces the ideal and initial ideal from the coordinates of a finite set of points" href="_affine__Points.html">affinePoints</a> -- produces the ideal and initial ideal from the coordinates of a finite set of points</span></li> <li><span><a title="computes ideal of point set by intersecting maximal ideals" href="_affine__Points__By__Intersection.html">affinePointsByIntersection</a> -- computes ideal of point set by intersecting maximal ideals</span></li> <li><span><a title="produces the matrix of values of the standard monomials on a set of points" href="_affine__Points__Mat.html">affinePointsMat</a> -- produces the matrix of values of the standard monomials on a set of points</span></li> <li><span><a title="The betti table of r points in Pn according to the minimal resolution conjecture" href="_expected__Betti.html">expectedBetti</a> -- The betti table of r points in Pn according to the minimal resolution conjecture</span></li> <li><span><a title="Min and max conceivable Betti tables for generic points" href="_min__Max__Resolution.html">minMaxResolution</a> -- Min and max conceivable Betti tables for generic points</span></li> <li><span><a title="linear part of the presentation of canonical module of points" href="_omega__Points.html">omegaPoints</a> -- linear part of the presentation of canonical module of points</span></li> <li><span><a title="make the ideal of a set of points" href="_points.html">points</a> -- make the ideal of a set of points</span></li> <li><span><a title="produces the ideal and initial ideal from the coordinates of a finite set of fat points" href="_projective__Fat__Points.html">projectiveFatPoints</a> -- produces the ideal and initial ideal from the coordinates of a finite set of fat points</span></li> <li><span><a title="computes ideal of fat points by intersecting powers of point ideals" href="_projective__Fat__Points__By__Intersection.html">projectiveFatPointsByIntersection</a> -- computes ideal of fat points by intersecting powers of point ideals</span></li> <li><span><a title="produces the ideal and initial ideal from the coordinates of a finite set of projective points" href="_projective__Points.html">projectivePoints</a> -- produces the ideal and initial ideal from the coordinates of a finite set of projective points</span></li> <li><span><a title="computes ideal of projective points by intersecting point ideals" href="_projective__Points__By__Intersection.html">projectivePointsByIntersection</a> -- computes ideal of projective points by intersecting point ideals</span></li> <li><span><a title="ideal of a random set of points" href="_random__Points.html">randomPoints</a> -- ideal of a random set of points</span></li> <li><span><a title="matrix of homogeneous coordinates of random points in projective space" href="_random__Points__Mat.html">randomPointsMat</a> -- matrix of homogeneous coordinates of random points in projective space</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>affineFatPoints(Matrix,List,Ring)</kbd> -- see <span><a title="produces the ideal and initial ideal from the coordinates of a finite set of fat points" href="_affine__Fat__Points.html">affineFatPoints</a> -- produces the ideal and initial ideal from the coordinates of a finite set of fat points</span></span></li> <li><span><kbd>affineFatPointsByIntersection(Matrix,List,Ring)</kbd> -- see <span><a title="computes ideal of fat points by intersecting powers of maximal ideals" href="_affine__Fat__Points__By__Intersection.html">affineFatPointsByIntersection</a> -- computes ideal of fat points by intersecting powers of maximal ideals</span></span></li> <li><span><kbd>affineMakeRingMaps(Matrix,Ring)</kbd> -- see <span><a title="evaluation on points" href="_affine__Make__Ring__Maps.html">affineMakeRingMaps</a> -- evaluation on points</span></span></li> <li><span><kbd>affinePoints(Matrix,Ring)</kbd> -- see <span><a title="produces the ideal and initial ideal from the coordinates of a finite set of points" href="_affine__Points.html">affinePoints</a> -- produces the ideal and initial ideal from the coordinates of a finite set of points</span></span></li> <li><span><kbd>affinePointsByIntersection(Matrix,Ring)</kbd> -- see <span><a title="computes ideal of point set by intersecting maximal ideals" href="_affine__Points__By__Intersection.html">affinePointsByIntersection</a> -- computes ideal of point set by intersecting maximal ideals</span></span></li> <li><span><kbd>affinePointsMat(Matrix,Ring)</kbd> -- see <span><a title="produces the matrix of values of the standard monomials on a set of points" href="_affine__Points__Mat.html">affinePointsMat</a> -- produces the matrix of values of the standard monomials on a set of points</span></span></li> <li><span><kbd>projectiveFatPoints(Matrix,List,Ring)</kbd> -- see <span><a title="produces the ideal and initial ideal from the coordinates of a finite set of fat points" href="_projective__Fat__Points.html">projectiveFatPoints</a> -- produces the ideal and initial ideal from the coordinates of a finite set of fat points</span></span></li> <li><span><kbd>projectiveFatPointsByIntersection(Matrix,List,Ring)</kbd> -- see <span><a title="computes ideal of fat points by intersecting powers of point ideals" href="_projective__Fat__Points__By__Intersection.html">projectiveFatPointsByIntersection</a> -- computes ideal of fat points by intersecting powers of point ideals</span></span></li> <li><span><kbd>projectivePoints(Matrix,Ring)</kbd> -- see <span><a title="produces the ideal and initial ideal from the coordinates of a finite set of projective points" href="_projective__Points.html">projectivePoints</a> -- produces the ideal and initial ideal from the coordinates of a finite set of projective points</span></span></li> <li><span><kbd>projectivePointsByIntersection(Matrix,Ring)</kbd> -- see <span><a title="computes ideal of projective points by intersecting point ideals" href="_projective__Points__By__Intersection.html">projectivePointsByIntersection</a> -- computes ideal of projective points by intersecting point ideals</span></span></li> <li><span><kbd>randomPointsMat(Ring,ZZ)</kbd> -- see <span><a title="matrix of homogeneous coordinates of random points in projective space" href="_random__Points__Mat.html">randomPointsMat</a> -- matrix of homogeneous coordinates of random points in projective space</span></span></li> </ul> </li> <li>Symbols <ul> <li><span><a title="Option to randomPointsMat." href="___All__Random.html">AllRandom</a> -- Option to randomPointsMat.</span></li> <li><span><a title="Option to projectivePoints." href="___Verify__Points.html">VerifyPoints</a> -- Option to projectivePoints.</span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="A package for making and studying points in affine and projective spaces" href="index.html">Points</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Points.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Points.m2:730:0</span>.</p> </div> </div> </div> </body> </html>