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<div><h1>InvolutiveBases -- Methods for Janet bases and Pommaret bases in Macaulay 2</h1>
<div class="single"><h2>Description</h2>
<div><em>InvolutiveBases</em> is a package which provides routines for dealing with Janet and Pommaret bases.<p>Janet bases can be constructed from given Gröbner bases. It can be checked whether a Janet basis is a Pommaret basis. Involutive reduction modulo a Janet basis can be performed. Syzygies and free resolutions can be computed using Janet bases. A convenient way to use this strategy is to use an optional argument for <a href="../../Macaulay2Doc/html/_resolution.html" title="projective resolution">resolution</a>, see <a href="___Involutive.html" title="compute a (usually non-minimal) resolution using involutive bases">Involutive</a>.</p>
<p>Some references:</p>
<ul><li>J. Apel, The theory of involutive divisions and an application to Hilbert function computations. J. Symb. Comp. 25(6), 1998, pp. 683-704.</li>
<li>V. P. Gerdt, Involutive Algorithms for Computing Gröbner Bases. In: Cojocaru, S. and Pfister, G. and Ufnarovski, V. (eds.), Computational Commutative and Non-Commutative Algebraic Geometry, NATO Science Series, IOS Press, pp. 199-225.</li>
<li>V. P. Gerdt and Y. A. Blinkov, Involutive bases of polynomial ideals. Minimal involutive bases. Mathematics and Computers in Simulation 45, 1998, pp. 519-541 resp. 543-560.</li>
<li>M. Janet, Lecons sur les systemes des equationes aux derivees partielles. Cahiers Scientifiques IV. Gauthiers-Villars, Paris, 1929.</li>
<li>J.-F. Pommaret, Partial Differential Equations and Group Theory. Kluwer Academic Publishers, 1994.</li>
<li>W. Plesken and D. Robertz, Janet's approach to presentations and resolutions for polynomials and linear pdes. Archiv der Mathematik 84(1), 2005, pp. 22-37.</li>
<li>D. Robertz, Janet Bases and Applications. In: Rosenkranz, M. and Wang, D. (eds.), Gröbner Bases in Symbolic Analysis, Radon Series on Computational and Applied Mathematics 2, de Gruyter, 2007, pp. 139-168.</li>
<li>W. M. Seiler, A Combinatorial Approach to Involution and delta-Regularity: I. Involutive Bases in Polynomial Algebras of Solvable Type. II. Structure Analysis of Polynomial Modules with Pommaret Bases. Preprints, arXiv:math/0208247 and arXiv:math/0208250.</li>
</ul>
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<div class="single"><h2>Author</h2>
<ul><li><div class="single"><a href="http://wwwb.math.rwth-aachen.de/~daniel/">Daniel Robertz</a><span> <<a href="mailto:daniel@momo.math.rwth-aachen.de">daniel@momo.math.rwth-aachen.de</a>></span></div>
</li>
</ul>
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<div class="single"><h2>Version</h2>
This documentation describes version <b>1.10</b> of InvolutiveBases.</div>
<div class="single"><h2>Source code</h2>
The source code from which this documentation is derived is in the file <a href="../../../../Macaulay2/InvolutiveBases.m2">InvolutiveBases.m2</a>.</div>
<div class="single"><h2>Exports</h2>
<ul><li><div class="single">Types<ul><li><span><a href="___Factor__Module__Basis.html" title="the class of all factor module bases">FactorModuleBasis</a> -- the class of all factor module bases</span></li>
<li><span><a href="___Involutive__Basis.html" title="the class of all involutive bases">InvolutiveBasis</a> -- the class of all involutive bases</span></li>
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<li><div class="single">Functions<ul><li><span><a href="_basis__Elements.html" title="extract the matrix of generators from an involutive basis or factor module basis">basisElements</a> -- extract the matrix of generators from an involutive basis or factor module basis</span></li>
<li><span><a href="_factor__Module__Basis.html" title="enumerate standard monomials">factorModuleBasis</a> -- enumerate standard monomials</span></li>
<li><span><a href="_inv__Noether__Normalization.html" title="Noether normalization">invNoetherNormalization</a> -- Noether normalization</span></li>
<li><span><a href="_inv__Reduce.html" title="compute normal form modulo involutive basis by involutive reduction">invReduce</a> -- compute normal form modulo involutive basis by involutive reduction</span></li>
<li><span><a href="_inv__Syzygies.html" title="compute involutive basis of syzygies">invSyzygies</a> -- compute involutive basis of syzygies</span></li>
<li><span><a href="_is__Pommaret__Basis.html" title="check whether or not a given Janet basis is also a Pommaret basis">isPommaretBasis</a> -- check whether or not a given Janet basis is also a Pommaret basis</span></li>
<li><span><a href="_janet__Basis.html" title="compute Janet basis for an ideal or a submodule of a free module">janetBasis</a> -- compute Janet basis for an ideal or a submodule of a free module</span></li>
<li><span><a href="_janet__Mult__Var.html" title="return table of multiplicative variables for given module elements as determined by Janet division">janetMultVar</a> -- return table of multiplicative variables for given module elements as determined by Janet division</span></li>
<li><span><a href="_janet__Resolution.html" title="construct a free resolution for a given ideal or module using Janet bases">janetResolution</a> -- construct a free resolution for a given ideal or module using Janet bases</span></li>
<li><span><a href="_mult__Var.html" title="extract the sets of multiplicative variables for each generator (in several contexts)">multVar</a> -- extract the sets of multiplicative variables for each generator (in several contexts)</span></li>
<li><span><a href="_pommaret__Mult__Var.html" title="return table of multiplicative variables for given module elements as determined by Pommaret division">pommaretMultVar</a> -- return table of multiplicative variables for given module elements as determined by Pommaret division</span></li>
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</li>
<li><div class="single">Methods<ul><li><span>invReduce(Matrix,InvolutiveBasis), see <span><a href="_inv__Reduce.html" title="compute normal form modulo involutive basis by involutive reduction">invReduce</a> -- compute normal form modulo involutive basis by involutive reduction</span></span></li>
<li><span>invReduce(RingElement,InvolutiveBasis), see <span><a href="_inv__Reduce.html" title="compute normal form modulo involutive basis by involutive reduction">invReduce</a> -- compute normal form modulo involutive basis by involutive reduction</span></span></li>
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<li><div class="single">Symbols<ul><li><span><a href="___Involutive.html" title="compute a (usually non-minimal) resolution using involutive bases">Involutive</a> -- compute a (usually non-minimal) resolution using involutive bases</span></li>
<li><span><a href="_mult__Vars.html" title="key in the cache table of a differential in a Janet resolution">multVars</a> -- key in the cache table of a differential in a Janet resolution</span></li>
<li><span><a href="___Permute__Variables.html" title="ensure that the last dim(I) var's are algebraically independent modulo I">PermuteVariables</a> -- ensure that the last dim(I) var's are algebraically independent modulo I</span></li>
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