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<!DOCTYPE html> <html lang="en"> <head> <title>Jets -- compute jets of various algebraic, geometric and combinatorial objects</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="compute jets of various algebraic, geometric and combinatorial objects" href="index.html">Jets</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> next | previous | <a href="_jets.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Jets -- compute jets of various algebraic, geometric and combinatorial objects</h1> <div> <h2>Description</h2> <div> <p>This package enables computations with jet functors. It introduces the <a title="compute the jets of an object" href="_jets.html">jets</a> method to compute jets of polynomial rings, ideals, quotients, ring homomorphisms, and varieties. The construction of jets follows an algebraic procedure discussed in several sources, including the first three references below.</p> <p>Additional features include an alternative algorithm to compute the radical of jets of monomial ideals, a function to construct jets of graphs, a method for principal components of jets, and an option for jets with "projective" gradings.</p> <h4>Version history:</h4> <ul> <li><b>1.1: </b>JSAG version.</li> <li><b>1.2: </b>Improved method for principal components. Added methods for invariants of principal components of monomial ideals.</li> </ul> </div> </div> <div> <h2>References</h2> <div> <p><a href="https://arxiv.org/abs/math/0612862">L. Ein and M. Mustaţă, Jet schemes and singularities.</a></p> <p><a href="https://arxiv.org/abs/math/0407113">P. Vojta, Jets via Hasse-Schmidt Derivations.</a></p> <p><a href="https://doi.org/10.1080/00927870500454927">R.A. Goward and K.E. Smith, The jet scheme of a monomial scheme.</a></p> <p><a href="https://arxiv.org/abs/2104.08933">F. Galetto, E. Helmick, and M. Walsh, Jet graphs.</a></p> <p><a href="https://arxiv.org/abs/2407.01836">F. Galetto, N. Iammarino, and T. Yu, Jets and principal components of monomial ideals, and very well-covered graphs</a></p> </div> </div> <div> <h3>Menu</h3> <h4>Package methods</h4> <ul> <li><span><a title="compute the jets of an object" href="_jets.html">jets</a> -- compute the jets of an object</span></li> <li><span><a title="canonical map between jets rings" href="_jets__Projection.html">jetsProjection</a> -- canonical map between jets rings</span></li> <li><span><a title="compute radicals of jets ideals" href="_jets__Radical.html">jetsRadical</a> -- compute radicals of jets ideals</span></li> <li><span><a title="compute principal component of jets" href="_principal__Component.html">principalComponent</a> -- compute principal component of jets</span></li> <li><span><a title="compute values of a lifting function" href="_lifting__Function.html">liftingFunction</a> -- compute values of a lifting function</span></li> </ul> <h4>Examples from the literature</h4> <ul> <li><span><a title="jets of monomial ideals" href="___Example_sp1.html">Example 1</a> -- jets of monomial ideals</span></li> <li><span><a title="jets of graphs" href="___Example_sp2.html">Example 2</a> -- jets of graphs</span></li> <li><span><a title="jets of determinantal varieties" href="___Example_sp3.html">Example 3</a> -- jets of determinantal varieties</span></li> <li><span><a title="invariants of principal jets of monomial ideals" href="___Example_sp4.html">Example 4</a> -- invariants of principal jets of monomial ideals</span></li> </ul> <h4>Technical information</h4> <ul> <li><span><a title="caching scheme for jets computations" href="___Storing_sp__Computations.html">Storing Computations</a> -- caching scheme for jets computations</span></li> </ul> </div> <div> <div> <div> <h2>Authors</h2> <ul> <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>Nicholas Iammarino<span> <<a href="mailto:nickiammarino%40gmail.com">nickiammarino@gmail.com</a>></span></li> </ul> </div> <div> <h2>Certification <img src="../../../../Macaulay2/Style/GoldStar.png" alt="a gold star"> </h2> <p>Version <b>1.1</b> of this package was accepted for publication in <a href="https://msp.org/jsag/2022/12-1/">volume 12</a> of <a href="https://msp.org/jsag/">The Journal of Software for Algebra and Geometry</a> on 20 October 2022, in the article <a href="https://msp.org/jsag/2022/12-1/p06.xhtml">Computing with jets</a> (DOI: <a href="https://doi.org/10.2140/jsag.2022.12.43">10.2140/jsag.2022.12.43</a>). That version can be obtained from <a href="https://msp.org/jsag/2022/12-1/jsag-v12-n1-x06-Jets.m2">the journal</a>.</p> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>1.2</b> of Jets, released <b>October 15, 2024</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{JetsSource, title = {{Jets: compute jets of various algebraic, geometric and combinatorial objects. Version~1.2}}, author = {Federico Galetto and Nicholas Iammarino}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } @article{JetsArticle, title = {{Computing with jets}}, author = {Federico Galetto and Nicholas Iammarino}, journal = {The Journal of Software for Algebra and Geometry}, volume = {12}, year = {2022}, } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Functions and commands <ul> <li><span><a title="compute the jets of an object" href="_jets.html">jets</a> -- compute the jets of an object</span></li> <li><span><a title="canonical map between jets rings" href="_jets__Projection.html">jetsProjection</a> -- canonical map between jets rings</span></li> <li><span><a title="compute radicals of jets ideals" href="_jets__Radical.html">jetsRadical</a> -- compute radicals of jets ideals</span></li> <li><span><a title="compute values of a lifting function" href="_lifting__Function.html">liftingFunction</a> -- compute values of a lifting function</span></li> <li><span><a title="arrange values of lifting function in a matrix" href="_lifting__Matrix.html">liftingMatrix</a> -- arrange values of lifting function in a matrix</span></li> <li><span><a title="compute principal component of jets" href="_principal__Component.html">principalComponent</a> -- compute principal component of jets</span></li> </ul> </li> <li>Methods <ul> <li><span><a title="the jets of a graph" href="_jets_lp__Z__Z_cm__Graph_rp.html">jets(ZZ,Graph)</a> -- the jets of a graph</span></li> <li><span><kbd>jets(ZZ,HyperGraph)</kbd> -- see <span><a title="the jets of a graph" href="_jets_lp__Z__Z_cm__Graph_rp.html">jets(ZZ,Graph)</a> -- the jets of a graph</span></span></li> <li><span><a title="compute jets of a an ideal in a polynomial ring" href="_jets_lp__Z__Z_cm__Ideal_rp.html">jets(ZZ,Ideal)</a> -- compute jets of a an ideal in a polynomial ring</span></li> <li><span><a title="compute jets of a polynomial ring" href="_jets_lp__Z__Z_cm__Polynomial__Ring_rp.html">jets(ZZ,PolynomialRing)</a> -- compute jets of a polynomial ring</span></li> <li><span><a title="the jets of an affine algebra" href="_jets_lp__Z__Z_cm__Quotient__Ring_rp.html">jets(ZZ,QuotientRing)</a> -- the jets of an affine algebra</span></li> <li><span><a title="the jets of a homomorphism of rings" href="_jets_lp__Z__Z_cm__Ring__Map_rp.html">jets(ZZ,RingMap)</a> -- the jets of a homomorphism of rings</span></li> <li><span><kbd>jetsProjection(ZZ,ZZ,PolynomialRing)</kbd> -- see <span><a title="canonical map between jets rings" href="_jets__Projection.html">jetsProjection</a> -- canonical map between jets rings</span></span></li> <li><span><kbd>jetsProjection(ZZ,ZZ,QuotientRing)</kbd> -- see <span><a title="canonical map between jets rings" href="_jets__Projection.html">jetsProjection</a> -- canonical map between jets rings</span></span></li> <li><span><kbd>jetsRadical(ZZ,Ideal)</kbd> -- see <span><a title="compute radicals of jets ideals" href="_jets__Radical.html">jetsRadical</a> -- compute radicals of jets ideals</span></span></li> <li><span><kbd>liftingFunction(ZZ,ZZ,ZZ)</kbd> -- see <span><a title="compute values of a lifting function" href="_lifting__Function.html">liftingFunction</a> -- compute values of a lifting function</span></span></li> <li><span><kbd>liftingMatrix(ZZ,ZZ,ZZ)</kbd> -- see <span><a title="arrange values of lifting function in a matrix" href="_lifting__Matrix.html">liftingMatrix</a> -- arrange values of lifting function in a matrix</span></span></li> <li><span><kbd>principalComponent(ZZ,Ideal)</kbd> -- see <span><a title="compute principal component of jets" href="_principal__Component.html">principalComponent</a> -- compute principal component of jets</span></span></li> </ul> </li> <li>Symbols <ul> <li><span><a title="hashtable key" href="_jet.html">jet</a> -- hashtable key</span></li> <li><span><a title="hashtable value" href="_jets__Base.html">jetsBase</a> -- hashtable value</span></li> <li><span><a title="hashtable key" href="_jets__Info.html">jetsInfo</a> -- hashtable key</span></li> <li><span><a title="hashtable value" href="_jets__Matrix.html">jetsMatrix</a> -- hashtable value</span></li> <li><span><a title="hashtable value" href="_jets__Max__Order.html">jetsMaxOrder</a> -- hashtable value</span></li> <li><span><a title="hashtable value" href="_jets__Ring.html">jetsRing</a> -- hashtable value</span></li> <li><span><a title="hashtable key" href="_projet.html">projet</a> -- hashtable key</span></li> <li><span><a title="option for principal components" href="___Saturate.html">Saturate</a> -- option for principal components</span></li> </ul> </li> <li>Other things <ul> <li><span><a title="scripted functor associated with jets" href="___J__J.html">JJ</a> -- scripted functor associated with jets</span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="compute jets of various algebraic, geometric and combinatorial objects" href="index.html">Jets</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Jets.m2</span>, with auxiliary files in <span class="tt">Jets/</span>.</p> </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>