<?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>ProjectiveHilbertPolynomial -- the class of all Hilbert polynomials</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_projective__Hilbert__Polynomial.html">next</a> | <a href="___Projective.html">previous</a> | <a href="_projective__Hilbert__Polynomial.html">forward</a> | <a href="___Projective.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>ProjectiveHilbertPolynomial -- the class of all Hilbert polynomials</h1>
<div class="single"><h2>Description</h2>
<div>For convenience, these polynomials are expressed in terms of the Hilbert polynomials of projective space.<p/>
The functions <a href="_degree.html" title="">degree</a> and <a href="_dim.html" title="compute the Krull dimension">dim</a> are designed so they correspond the degree and dimension of the algebraic variety that may have been used to produce the Hilbert polynomial.<table class="examples"><tr><td><pre>i1 : Z = Proj(QQ[x_0..x_12]/(x_0^3+x_12^3))
o1 = Z
o1 : ProjectiveVariety</pre>
</td></tr>
<tr><td><pre>i2 : hilbertPolynomial Z
o2 = P - 3*P + 3*P
9 10 11
o2 : ProjectiveHilbertPolynomial</pre>
</td></tr>
</table>
</div>
</div>
<div class="waystouse"><h2>Functions and methods returning a projective Hilbert polynomial :</h2>
<ul><li><span>ZZ * ProjectiveHilbertPolynomial, see <span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></span></li>
<li><span>ProjectiveHilbertPolynomial + ProjectiveHilbertPolynomial, see <span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></span></li>
<li><span>- ProjectiveHilbertPolynomial, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span>ProjectiveHilbertPolynomial - ProjectiveHilbertPolynomial, see <span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></span></li>
<li><span><a href="_diff_lp__Projective__Hilbert__Polynomial_rp.html" title="">diff(ProjectiveHilbertPolynomial)</a></span></li>
<li><span><a href="_diff_lp__Projective__Hilbert__Polynomial_cm__Z__Z_rp.html" title="">diff(ProjectiveHilbertPolynomial,ZZ)</a></span></li>
<li><span><a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a> -- compute the Hilbert polynomial</span></li>
<li><span><a href="_hilbert__Polynomial_lp__Module_rp.html" title="compute the Hilbert polynomial of the module">hilbertPolynomial(Module)</a> -- compute the Hilbert polynomial of the module</span></li>
<li><span><a href="_hilbert__Polynomial_lp__Projective__Variety_rp.html" title="compute the Hilbert polynomial of the projective variety">hilbertPolynomial(ProjectiveVariety)</a> -- compute the Hilbert polynomial of the projective variety</span></li>
<li><span><a href="_hilbert__Polynomial_lp__Ring_rp.html" title="compute the Hilbert polynomial of the ring">hilbertPolynomial(Ring)</a> -- compute the Hilbert polynomial of the ring</span></li>
<li><span>projectiveHilbertPolynomial(ZZ), see <span><a href="_projective__Hilbert__Polynomial.html" title="Hilbert polynomial of projective space">projectiveHilbertPolynomial</a> -- Hilbert polynomial of projective space</span></span></li>
<li><span>projectiveHilbertPolynomial(ZZ,ZZ), see <span><a href="_projective__Hilbert__Polynomial.html" title="Hilbert polynomial of projective space">projectiveHilbertPolynomial</a> -- Hilbert polynomial of projective space</span></span></li>
</ul>
<h2>Methods that use a projective Hilbert polynomial :</h2>
<ul><li><span>ProjectiveHilbertPolynomial == ProjectiveHilbertPolynomial, see <span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></span></li>
<li><span><a href="_degree_lp__Projective__Hilbert__Polynomial_rp.html" title="">degree(ProjectiveHilbertPolynomial)</a></span></li>
<li><span><a href="_dim_lp__Projective__Hilbert__Polynomial_rp.html" title="the degree of the Hilbert polynomial">dim(ProjectiveHilbertPolynomial)</a> -- the degree of the Hilbert polynomial</span></li>
<li><span><a href="_euler_lp__Projective__Hilbert__Polynomial_rp.html" title="constant term of the Hilbert polynomial">euler(ProjectiveHilbertPolynomial)</a> -- constant term of the Hilbert polynomial</span></li>
<li><span><a href="_hilbert__Series_lp__Projective__Hilbert__Polynomial_rp.html" title="compute the Hilbert series of a projective Hilbert polynomial">hilbertSeries(ProjectiveHilbertPolynomial)</a> -- compute the Hilbert series of a projective Hilbert polynomial</span></li>
<li><span><a href="___Projective__Hilbert__Polynomial_sp__Z__Z.html" title="value of polynomial">ProjectiveHilbertPolynomial ZZ</a> -- value of polynomial</span></li>
</ul>
</div>
<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Projective__Hilbert__Polynomial.html" title="the class of all Hilbert polynomials">ProjectiveHilbertPolynomial</a> is <span>a <a href="___Type.html">type</a></span>, with ancestor classes <a href="___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="___Thing.html" title="the class of all things">Thing</a>.</p>
</div>
</div>
</body>
</html>
