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<!DOCTYPE html> <html lang="en"> <head> <title>Normaliz -- an interface to use Normaliz in Macaulay 2</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="an interface to use Normaliz in Macaulay 2" href="index.html">Normaliz</a> :: <a title="an interface to use Normaliz in Macaulay 2" href="index.html">Normaliz</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="_all__Computations.html">next</a> | previous | <a href="_all__Computations.html">forward</a> | backward | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>Normaliz -- an interface to use Normaliz in Macaulay 2</h1> <div> <h2>Description</h2> The package <em>Normaliz</em> provides an interface for the use of <span class="tt">Normaliz 2.8</span> within Macaulay 2. <p></p> The program <span class="tt">Normaliz 2.8</span> (referred to as <span class="tt">Normaliz</span> in the following) is mainly a tool for computing the Hilbert bases and enumerative invariants of rational cones. Several additional data can be computed. It is included in the Macaulay 2 distribution. For more details on the program, see <a href="http://www.math.uos.de/normaliz/">http://www.math.uos.de/normaliz/</a>. For the theory of affine semigroups and the notions of commutative algebra we refer to W. Bruns and J. Gubeladze, <em>Polytopes, rings and K-theory.</em> Springer 2009. <p></p> For algorithms see: <ul> <li>W. Bruns and R. Koch, <em>Computing the integral closure of an affine semigroup. </em>Univ. Iagiell. Acta Math. 39, (2001), 59-70</li> <li>W. Bruns and B. Ichim <em>Normaliz: Algorithms for affine monoids and rational cones,</em> J. Algebra (2010), available at <a href="http://dx.doi.org/10.1016/j.jalgebra.2010.01.031">http://dx.doi.org/10.1016/j.jalgebra.2010.01.031</a></li> <li>W. Bruns, B. Ichim and C.Soeger <em>The power of pyramid decomposition in Normaliz,</em> arXiv:1206.1916v1, available at <a href="http://arxiv.org/abs/1206.1916v1">http://arxiv.org/abs/1206.1916v1</a></li> </ul> <p></p> Using <span class="tt">Normaliz</span> one may for example compute the following: <ul> <li><a href="___The_sp__Hilbert_spbasis_spand_spthe_spsupport_sphyperplanes_spof_spa_sprational_spcone..html">The Hilbert basis and the support hyperplanes of a rational cone.</a></li> <li><a href="___The_splattice_sppoints_cm_sp__Ehrhart_spseries_spand_spthe_spsupport_sphyperplanes_spof_span_spintegral_sppolytope..html">The lattice points, Ehrhart series and the support hyperplanes of an integral polytope.</a></li> <li><span><a href="___The_spgenerators_spof_spthe_spintegral_spclosure_spof_spthe_sp__Rees_spalgebra_spof_spa_spmonomial_spideal..html">The generators of the integral closure of the Rees algebra of a monomial ideal.</a></span></li> </ul> <p></p> If the associated semigroup or corresponding semigroup algebra is graded, then one may also compute the Hilbert series and Hilbert (quasi)polynomial of the semigroup. <p></p> The package gives direct access to <span class="tt">Normaliz</span>. The exchange of data between <span class="tt">Normaliz</span> and Macaulay 2 is via files. These files are automatically created and erased behind the scenes. As long as one wants to use only the ring-theoretic functions there is no need for file management. The key function for the direct use of <span class="tt">Normaliz</span> is <a title="calls Normaliz" href="_normaliz.html">normaliz</a>, which calls the program <span class="tt">Normaliz</span>. To handle the in- and output one can use the functions <a title="creates an input file for Normaliz" href="_write__Nmz__Data.html">writeNmzData</a> and <a title="reads an output file of Normaliz containing one matrix" href="_read__Nmz__Data.html">readNmzData</a>, to set the options for the program <a title="sets a command line option for Normaliz" href="_set__Nmz__Option.html">setNmzOption</a>. The output files are explained in <a href="_output_spfiles_spwritten_spby_sp__Normaliz.html">output files written by Normaliz</a>. <p></p> If you want to keep the results of the computations by <span class="tt">Normaliz</span> (i.e. the files written by the program), the package offers several methods for this purpose, see <a href="___Keeping_spresults_spof_spthe_spcomputation_spby_sp__Normaliz.html">Keeping results of the computation by Normaliz</a> for an example how to do this. <p></p> The package introduces two new classes <a title="class of monomial subalgebras" href="___Monomial__Subalgebra.html">MonomialSubalgebra</a> and <a title="class of rational cones" href="___Rational__Cone.html">RationalCone</a>. <p></p> The package provides four top level functions that aim directly at algebraic objects: <ul> <li><span><a title="normalization of a toric ring" href="_normal__Toric__Ring.html">normalToricRing</a> -- normalization of a toric ring</span></li> <li><span><a title="integral closure of a toric ring" href="_intcl__Toric__Ring.html">intclToricRing</a> -- integral closure of a toric ring</span></li> <li><span><a title="normalization of Rees algebra" href="_intcl__Mon__Ideal.html">intclMonIdeal</a> -- normalization of Rees algebra</span></li> <li><span><a title="Ehrhart ring" href="_ehrhart__Ring.html">ehrhartRing</a> -- Ehrhart ring</span></li> </ul> <p></p> The package offers the following additional functions: <ul> <li><span><a title="ring of invariants of torus action" href="_torus__Invariants.html">torusInvariants</a> -- ring of invariants of torus action</span></li> <li><span><a title="ring of invariants of a finite group action" href="_finite__Diag__Invariants.html">finiteDiagInvariants</a> -- ring of invariants of a finite group action</span></li> <li><span><a title="ring of invariants of a diagonalizable group action" href="_diag__Invariants.html">diagInvariants</a> -- ring of invariants of a diagonalizable group action</span></li> <li><span><a title="intersection of ring of valuations" href="_intersection__Val__Rings.html">intersectionValRings</a> -- intersection of ring of valuations</span></li> <li><span><a title="intersection of valuation ideals" href="_intersection__Val__Ring__Ideals.html">intersectionValRingIdeals</a> -- intersection of valuation ideals</span></li> </ul> </div> <div> <div> <div> <h2>Authors</h2> <ul> <li>Gesa Kaempf<span> <<a href="mailto:gkaempf%40uni-osnabrueck.de">gkaempf@uni-osnabrueck.de</a>></span></li> <li>Christof Soeger<span> <<a href="mailto:csoeger%40uni-osnabrueck.de">csoeger@uni-osnabrueck.de</a>></span></li> </ul> </div> <div> <h2>Certification <img src="../../../../Macaulay2/Style/GoldStar.png" alt="a gold star"> </h2> <p>Version <b>2.0</b> of this package was accepted for publication in <a href="https://msp.org/jsag/2010/2-1/">volume 2</a> of <a href="https://msp.org/jsag/">The Journal of Software for Algebra and Geometry: Macaulay2</a> on 2010-08-08, in the article <a href="https://msp.org/jsag/2010/2-1/p04.xhtml">A Macaulay2 interface for Normaliz</a> (DOI: <a href="https://doi.org/10.2140/jsag.2010.2.15">10.2140/jsag.2010.2.15</a>). That version can be obtained from <a href="https://msp.org/jsag/2010/2-1/jsag-v2-n1-x04-code.zip">the journal</a>.</p> </div> <div> <h2>Version</h2> <p>This documentation describes version <b>2.6</b> of Normaliz, released <b>February 4, 2023</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{NormalizSource, title = {{Normaliz: interface to Normaliz in \emph{Macaulay2}. Version~2.6}}, author = {Gesa Kaempf and Christof Soeger}, howpublished = {A \emph{Macaulay2} package available at \url{https://github.com/Macaulay2/M2/tree/stable/M2/Macaulay2/packages}} } @article{NormalizArticle, title = {{A \emph{Macaulay2} interface for Normaliz}}, author = {Gesa Kaempf and Christof Soeger}, journal = {The Journal of Software for Algebra and Geometry: Macaulay2}, volume = {2}, year = {2010}, } </code></pre> </td> </tr> </table> </div> <div> <h2>Exports</h2> <div class="exports"> <ul> <li>Types <ul> <li><span><a title="class of monomial subalgebras" href="___Monomial__Subalgebra.html">MonomialSubalgebra</a> -- class of monomial subalgebras</span></li> <li><span><a title="class of rational cones" href="___Rational__Cone.html">RationalCone</a> -- class of rational cones</span></li> </ul> </li> <li>Functions and commands <ul> <li><span><a title="creates a monomial subalgebra" href="_create__Monomial__Subalgebra.html">createMonomialSubalgebra</a> -- creates a monomial subalgebra</span></li> <li><span><a title="ring of invariants of a diagonalizable group action" href="_diag__Invariants.html">diagInvariants</a> -- ring of invariants of a diagonalizable group action</span></li> <li><span><a title="Ehrhart ring" href="_ehrhart__Ring.html">ehrhartRing</a> -- Ehrhart ring</span></li> <li><span><a title="ring of invariants of a finite group action" href="_finite__Diag__Invariants.html">finiteDiagInvariants</a> -- ring of invariants of a finite group action</span></li> <li><span><a title="returns the numerical invariants computed" href="_get__Num__Invs.html">getNumInvs</a> -- returns the numerical invariants computed</span></li> <li><span><a title="normalization of Rees algebra" href="_intcl__Mon__Ideal.html">intclMonIdeal</a> -- normalization of Rees algebra</span></li> <li><span><a title="integral closure of a toric ring" href="_intcl__Toric__Ring.html">intclToricRing</a> -- integral closure of a toric ring</span></li> <li><span><a title="intersection of valuation ideals" href="_intersection__Val__Ring__Ideals.html">intersectionValRingIdeals</a> -- intersection of valuation ideals</span></li> <li><span><a title="intersection of ring of valuations" href="_intersection__Val__Rings.html">intersectionValRings</a> -- intersection of ring of valuations</span></li> <li><span><a title="calls Normaliz" href="_normaliz.html">normaliz</a> -- calls Normaliz</span></li> <li><span><a title="normalization of a toric ring" href="_normal__Toric__Ring.html">normalToricRing</a> -- normalization of a toric ring</span></li> <li><span><a title="reads an output file of Normaliz containing several matrices" href="_read__Multiple__Nmz__Data.html">readMultipleNmzData</a> -- reads an output file of Normaliz containing several matrices</span></li> <li><span><a title="reads an output file of Normaliz containing one matrix" href="_read__Nmz__Data.html">readNmzData</a> -- reads an output file of Normaliz containing one matrix</span></li> <li><span><a title="removes the files created by Normaliz" href="_rm__Nmz__Files.html">rmNmzFiles</a> -- removes the files created by Normaliz</span></li> <li><span><a title="sets a command line option for Normaliz" href="_set__Nmz__Option.html">setNmzOption</a> -- sets a command line option for Normaliz</span></li> <li><span><a title="prints the enabled options" href="_show__Nmz__Options.html">showNmzOptions</a> -- prints the enabled options</span></li> <li><span><a title="ring of invariants of torus action" href="_torus__Invariants.html">torusInvariants</a> -- ring of invariants of torus action</span></li> <li><span><a title="creates an input file for Normaliz" href="_write__Nmz__Data.html">writeNmzData</a> -- creates an input file for Normaliz</span></li> </ul> </li> <li>Methods <ul> <li><span><kbd>createMonomialSubalgebra(List)</kbd> -- see <span><a title="creates a monomial subalgebra" href="_create__Monomial__Subalgebra.html">createMonomialSubalgebra</a> -- creates a monomial subalgebra</span></span></li> <li><span><kbd>diagInvariants(Matrix,Matrix,Ring)</kbd> -- see <span><a title="ring of invariants of a diagonalizable group action" href="_diag__Invariants.html">diagInvariants</a> -- ring of invariants of a diagonalizable group action</span></span></li> <li><span><a title="Ehrhart ring" href="_ehrhart__Ring_lp__List_rp.html">ehrhartRing(List)</a> -- Ehrhart ring</span></li> <li><span><a title="Ehrhart ring" href="_ehrhart__Ring_lp__List_cm__Ring__Element_rp.html">ehrhartRing(List,RingElement)</a> -- Ehrhart ring</span></li> <li><span><kbd>finiteDiagInvariants(Matrix,Ring)</kbd> -- see <span><a title="ring of invariants of a finite group action" href="_finite__Diag__Invariants.html">finiteDiagInvariants</a> -- ring of invariants of a finite group action</span></span></li> <li><span><a title="generators of a monomial subalgebra" href="_generators_lp__Monomial__Subalgebra_rp.html">generators(MonomialSubalgebra)</a> -- generators of a monomial subalgebra</span></li> <li><span><a title="normalization of Rees algebra" href="_intcl__Mon__Ideal_lp__Ideal_rp.html">intclMonIdeal(Ideal)</a> -- normalization of Rees algebra</span></li> <li><span><a title="normalization of Rees algebra" href="_intcl__Mon__Ideal_lp__Ideal_cm__Ring__Element_rp.html">intclMonIdeal(Ideal,RingElement)</a> -- normalization of Rees algebra</span></li> <li><span><a title="integral closure of a toric ring" href="_intcl__Toric__Ring_lp__List_rp.html">intclToricRing(List)</a> -- integral closure of a toric ring</span></li> <li><span><a title="integral closure of a toric ring" href="_intcl__Toric__Ring_lp__Monomial__Subalgebra_rp.html">intclToricRing(MonomialSubalgebra)</a> -- integral closure of a toric ring</span></li> <li><span><kbd>intersectionValRingIdeals(Matrix,Ring)</kbd> -- see <span><a title="intersection of valuation ideals" href="_intersection__Val__Ring__Ideals.html">intersectionValRingIdeals</a> -- intersection of valuation ideals</span></span></li> <li><span><kbd>intersectionValRings(Matrix,Ring)</kbd> -- see <span><a title="intersection of ring of valuations" href="_intersection__Val__Rings.html">intersectionValRings</a> -- intersection of ring of valuations</span></span></li> <li><span><a title="calls Normaliz with several input matrices" href="_normaliz_lp__List_rp.html">normaliz(List)</a> -- calls Normaliz with several input matrices</span></li> <li><span><a title="calls Normaliz" href="_normaliz_lp__Matrix_cm__String_rp.html">normaliz(Matrix,String)</a> -- calls Normaliz</span></li> <li><span><a title="normalization of a toric ring given by a binomial ideal" href="_normal__Toric__Ring_lp__Ideal_cm__Thing_rp.html">normalToricRing(Ideal,Thing)</a> -- normalization of a toric ring given by a binomial ideal</span></li> <li><span><a title="normalization of a toric ring" href="_normal__Toric__Ring_lp__List_rp.html">normalToricRing(List)</a> -- normalization of a toric ring</span></li> <li><span><a title="normalization of a toric ring" href="_normal__Toric__Ring_lp__Monomial__Subalgebra_rp.html">normalToricRing(MonomialSubalgebra)</a> -- normalization of a toric ring</span></li> <li><span><a title="number of generators of a monomial subalgebra" href="_numgens_lp__Monomial__Subalgebra_rp.html">numgens(MonomialSubalgebra)</a> -- number of generators of a monomial subalgebra</span></li> <li><span><kbd>readMultipleNmzData(String)</kbd> -- see <span><a title="reads an output file of Normaliz containing several matrices" href="_read__Multiple__Nmz__Data.html">readMultipleNmzData</a> -- reads an output file of Normaliz containing several matrices</span></span></li> <li><span><kbd>readNmzData(String)</kbd> -- see <span><a title="reads an output file of Normaliz containing one matrix" href="_read__Nmz__Data.html">readNmzData</a> -- reads an output file of Normaliz containing one matrix</span></span></li> <li><span><a title="surrounding ring of a monomial subalgebra" href="_ring_lp__Monomial__Subalgebra_rp.html">ring(MonomialSubalgebra)</a> -- surrounding ring of a monomial subalgebra</span></li> <li><span><kbd>setNmzOption(String,Boolean)</kbd> -- see <span><a title="sets a command line option for Normaliz" href="_set__Nmz__Option.html">setNmzOption</a> -- sets a command line option for Normaliz</span></span></li> <li><span><kbd>torusInvariants(Matrix,Ring)</kbd> -- see <span><a title="ring of invariants of torus action" href="_torus__Invariants.html">torusInvariants</a> -- ring of invariants of torus action</span></span></li> <li><span><a title="row matrix of the generators of a monomial subalgebra" href="_vars_lp__Monomial__Subalgebra_rp.html">vars(MonomialSubalgebra)</a> -- row matrix of the generators of a monomial subalgebra</span></li> <li><span><a title="creates an input file for Normaliz with several matrices" href="_write__Nmz__Data_lp__List_rp.html">writeNmzData(List)</a> -- creates an input file for Normaliz with several matrices</span></li> <li><span><a title="creates an input file for Normaliz with one matrix" href="_write__Nmz__Data_lp__Matrix_cm__String_rp.html">writeNmzData(Matrix,String)</a> -- creates an input file for Normaliz with one matrix</span></li> </ul> </li> <li>Symbols <ul> <li><span><a href="_all__Computations.html">allComputations</a></span></li> <li><span><a href="_grading.html">grading</a></span></li> </ul> </li> <li>Mutable symbols <ul> <li><span><a title="global variable, the path where Normaliz stores its files" href="_nmz__Data__Path.html">nmzDataPath</a> -- global variable, the path where Normaliz stores its files</span></li> <li><span><a title="global variable holding the filename" href="_nmz__Filename.html">nmzFilename</a> -- global variable holding the filename</span></li> <li><span><a title="global variable holding the number of threads" href="_nmz__Number__Threads.html">nmzNumberThreads</a> -- global variable holding the number of threads</span></li> <li><span><a title="global variable holding the Normaliz version" href="_nmz__Version.html">nmzVersion</a> -- global variable holding the Normaliz version</span></li> </ul> </li> <li>Other things <ul> <li><span><a title="global variable, the path where Normaliz stores its files" href="_nmz__Data__Path.html">nmzDataPath</a> -- global variable, the path where Normaliz stores its files</span></li> <li><span><a title="global variable holding the filename" href="_nmz__Filename.html">nmzFilename</a> -- global variable holding the filename</span></li> <li><span><a title="global variable holding the number of threads" href="_nmz__Number__Threads.html">nmzNumberThreads</a> -- global variable holding the number of threads</span></li> <li><span><a title="global variable holding the Normaliz version" href="_nmz__Version.html">nmzVersion</a> -- global variable holding the Normaliz version</span></li> </ul> </li> </ul> </div> </div> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="an interface to use Normaliz in Macaulay 2" href="index.html">Normaliz</a> is <span>a <a title="the class of all packages" href="../../Macaulay2Doc/html/___Package.html">package</a></span>, defined in <span class="tt">Normaliz.m2</span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Normaliz.m2:682:0</span>.</p> </div> </div> </div> </body> </html>
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