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<!DOCTYPE html> <html lang="en"> <head> <title>Example 1 -- jets of monomial ideals</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="compute jets of various algebraic, geometric and combinatorial objects" href="index.html">Jets</a> » <a title="jets of monomial ideals" href="___Example_sp1.html">Example 1</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="___Example_sp2.html">next</a> | <a href="_lifting__Function.html">previous</a> | <a href="___Example_sp2.html">forward</a> | <a href="_lifting__Matrix.html">backward</a> | <a href="index.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Example 1 -- jets of monomial ideals</h1> <div> <div> <p>As observed in <a href="https://doi.org/10.1080/00927870500454927">R.A. Goward and K.E. Smith, The jet scheme of a monomial scheme</a> [GS06], the ideal of jets of a monomial ideal is typically not a monomial ideal.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : R = QQ[x,y,z] o1 = R o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : I = ideal(x*y*z) o2 = ideal(x*y*z) o2 : Ideal of R</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : J2I = jets(2,I) o3 = ideal (y0*z0*x2 + x0*z0*y2 + x0*y0*z2 + z0*x1*y1 + y0*x1*z1 + x0*y1*z1, ------------------------------------------------------------------------ y0*z0*x1 + x0*z0*y1 + x0*y0*z1, x0*y0*z0) o3 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]</code></pre> </td> </tr> </table> <div> <p>However, by [GS06, Theorem 3.1], the radical is always a (squarefree) monomial ideal. In fact, the proof of [GS06, Theorem 3.2] shows that the radical is generated by the individual terms in the generators of the ideal of jets. This observation provides an alternative algorithm for computing radicals of jets of monomial ideals, which can be faster than the default radical computation in Macaulay2.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i4 : elapsedTime jetsRadical(2,I) -- .00270135s elapsed o4 = ideal (y0*z0*x2, x0*z0*y2, x0*y0*z2, z0*x1*y1, y0*x1*z1, x0*y1*z1, ------------------------------------------------------------------------ y0*z0*x1, x0*z0*y1, x0*y0*z1, x0*y0*z0) o4 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i5 : elapsedTime radical J2I -- .318314s elapsed o5 = ideal (x0*y0*z0, x0*y0*z1, x0*z0*y1, y0*z0*x1, x0*y1*z1, y0*x1*z1, ------------------------------------------------------------------------ z0*x1*y1, x0*y0*z2, x0*z0*y2, y0*z0*x2) o5 : Ideal of QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2]</code></pre> </td> </tr> </table> <div> <p>For a monomial hypersurface, [GS06, Theorem 3.2] describes the minimal primes of the ideal of jets. Moreover, the main theorem in <a href="https://arxiv.org/abs/math/0607638">C. Yuen, Multiplicity of jet schemes of monomial schemes</a> counts the multiplicity of the jet scheme of a monomial hypersurface along its minimal primes (see also <a href="https://doi.org/10.1080/00927870701512168">C. Yuen, The multiplicity of jet schemes of a simple normal crossing divisor</a>). We compute the minimal primes, then use the <a title="Localizations of polynomial rings at prime ideals" href="../../LocalRings/html/index.html">LocalRings</a> package to compute their multiplicities in the second jet scheme of the example above. Note that we need to flatten the polynomial ring of jets because the <span class="tt">LocalRings</span> package does not allow towers of rings.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i6 : P = minimalPrimes J2I o6 = {ideal (z0, y0, x0), ideal (z0, y0, z1), ideal (z0, y0, y1), ideal (z0, ------------------------------------------------------------------------ x0, z1), ideal (z0, x0, x1), ideal (z0, z1, z2), ideal (y0, x0, y1), ------------------------------------------------------------------------ ideal (y0, x0, x1), ideal (y0, y1, y2), ideal (x0, x1, x2)} o6 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : (A,f) = flattenRing ring J2I o7 = (A, map (A, QQ[x0, y0, z0][x1, y1, z1][x2, y2, z2], {x2, y2, z2, x1, y1, ------------------------------------------------------------------------ z1, x0, y0, z0})) o7 : Sequence</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : needsPackage "LocalRings" o8 = LocalRings o8 : Package</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i9 : M = cokernel gens f J2I o9 = cokernel | x2y0z0+y2x0z0+z2x0y0+x1y1z0+x1z1y0+y1z1x0 x1y0z0+y1x0z0+z1x0y0 x0y0z0 | 1 o9 : A-module, quotient of A</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i10 : mult = for p in P list ( Rp := localRing(A,f p); length(M ** Rp) );</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i11 : netList(pack(4,mingle{P,mult}),HorizontalSpace=>1) +--------------------+---+--------------------+---+ o11 = | ideal (z0, y0, x0) | 6 | ideal (z0, y0, z1) | 3 | +--------------------+---+--------------------+---+ | ideal (z0, y0, y1) | 3 | ideal (z0, x0, z1) | 3 | +--------------------+---+--------------------+---+ | ideal (z0, x0, x1) | 3 | ideal (z0, z1, z2) | 1 | +--------------------+---+--------------------+---+ | ideal (y0, x0, y1) | 3 | ideal (y0, x0, x1) | 3 | +--------------------+---+--------------------+---+ | ideal (y0, y1, y2) | 1 | ideal (x0, x1, x2) | 1 | +--------------------+---+--------------------+---+</code></pre> </td> </tr> </table> </div> <div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Jets.m2:1666:0</span>.</p> </div> </div> </div> </body> </html>