<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- --> <!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html --> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <head><title>isHF -- is a finite list a Hilbert function of a polynomial ring mod a homogeneous ideal</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="_is__Lex__Ideal.html">next</a> | <a href="_is__C__M.html">previous</a> | <a href="_is__Lex__Ideal.html">forward</a> | <a href="_is__C__M.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div> </td> </tr> </table> <hr/> <div><h1>isHF -- is a finite list a Hilbert function of a polynomial ring mod a homogeneous ideal</h1> <div class="single"><h2>Synopsis</h2> <ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>b=isHF L</tt></div> </dd></dl> </div> </li> <li><div class="single">Inputs:<ul><li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a finite list of integers</span></li> </ul> </div> </li> <li><div class="single">Outputs:<ul><li><span><tt>b</tt>, <span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <tt>true</tt> if <tt>L</tt> is a Hilbert function of a polynomial ring modulo a homogeneous ideal and <tt>false</tt> otherwise</span></li> </ul> </div> </li> </ul> </div> <div class="single"><h2>Description</h2> <div><div>Macaulay’s Theorem characterizes the sequences of integers that occur as the Hilbert function of a polynomial ring modulo a homogeneous ideal. <tt>isHF</tt> checks that the input is a list of integers and that the first entry of the list is 1, and then it checks Macaulay’s bound in each degree, using <a href="_macaulay__Bound.html" title="the bound on the growth of a Hilbert function from Macaulay's Theorem">macaulayBound</a>. The function returns <tt>true</tt> if the sequence of numbers in the list satisfies the conditions of Macaulay’s Theorem and <tt>false</tt> otherwise.</div> <table class="examples"><tr><td><pre>i1 : isHF({1,3,6,7,5,3}) o1 = true</pre> </td></tr> <tr><td><pre>i2 : isHF({2,3,4,3,2}) --doesn't start with a 1 in degree 0 o2 = false</pre> </td></tr> <tr><td><pre>i3 : isHF({1,3,6,8,14,3}) --growth from 8 to 14 is too high o3 = false</pre> </td></tr> </table> </div> </div> <div class="single"><h2>See also</h2> <ul><li><span><a href="_macaulay__Rep.html" title="the Macaulay representation of an integer">macaulayRep</a> -- the Macaulay representation of an integer</span></li> <li><span><a href="_macaulay__Bound.html" title="the bound on the growth of a Hilbert function from Macaulay's Theorem">macaulayBound</a> -- the bound on the growth of a Hilbert function from Macaulay's Theorem</span></li> </ul> </div> <div class="waystouse"><h2>Ways to use <tt>isHF</tt> :</h2> <ul><li>isHF(List)</li> </ul> </div> </div> </body> </html>