<?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>Singular Book 1.8.6 -- Zariski closure of the image</title> <link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/> </head> <body> <table class="buttons"> <tr> <td><div><a href="___Singular_sp__Book_sp1.8.7.html">next</a> | <a href="___Singular_sp__Book_sp1.8.4.html">previous</a> | <a href="___Singular_sp__Book_sp1.8.7.html">forward</a> | <a href="___Singular_sp__Book_sp1.8.4.html">backward</a> | <a href="___M2__Singular__Book.html">up</a> | <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> <div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_basic_spcommutative_spalgebra.html" title="">basic commutative algebra</a> > <a href="___M2__Singular__Book.html" title="Macaulay2 examples for the Singular book">M2SingularBook</a> > <a href="___Singular_sp__Book_sp1.8.6.html" title="Zariski closure of the image">Singular Book 1.8.6</a></div> <hr/> <div><h1>Singular Book 1.8.6 -- Zariski closure of the image</h1> <div>We compute an implicit equation for the surface defined parametrically by the map <i>f : A<sup>2</sup> → A<sup>3</sup>, (u,v) → (uv,uv<sup>2</sup>,u<sup>2</sup>)</i>.<table class="examples"><tr><td><pre>i1 : loadPackage "Elimination";</pre> </td></tr> <tr><td><pre>i2 : A = QQ[u,v,x,y,z];</pre> </td></tr> <tr><td><pre>i3 : I = ideal "x-uv,y-uv2,z-u2" 2 2 o3 = ideal (- u*v + x, - u*v + y, - u + z) o3 : Ideal of A</pre> </td></tr> <tr><td><pre>i4 : eliminate(I,{u,v}) 4 2 o4 = ideal(x - y z) o4 : Ideal of A</pre> </td></tr> </table> This ideal defines the closure of the map <i>f</i>, the Whitney umbrella.<p>Alternatively, we could take the coimage of the ring homomorphism <tt>g</tt> corresponding to <tt>f</tt>.</p> <table class="examples"><tr><td><pre>i5 : g = map(QQ[u,v],QQ[x,y,z],{x => u*v, y => u*v^2, z => u^2}) 2 2 o5 = map(QQ[u, v],QQ[x, y, z],{u*v, u*v , u }) o5 : RingMap QQ[u, v] <--- QQ[x, y, z]</pre> </td></tr> <tr><td><pre>i6 : coimage g QQ[x, y, z] o6 = ----------- 4 2 x - y z o6 : QuotientRing</pre> </td></tr> </table> </div> </div> </body> </html>