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Mathematics of Computation

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How to compute the Stanley depth of a module

Authors: Bogdan Ichim, Lukas Katthän and Julio José Moyano-Fernández
Journal: Math. Comp. 86 (2017), 455-472
MSC (2010): Primary 05A18, 05E40; Secondary 16W50
Published electronically: April 13, 2016
MathSciNet review: 3557807
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Abstract: In this paper we introduce an algorithm for computing the Stanley depth of a finitely generated multigraded module $ M$ over the polynomial ring $ \mathbb{K}[X_1, \ldots , X_n]$. As an application, we give an example of a module whose Stanley depth is strictly greater than the depth of its syzygy module. In particular, we obtain complete answers for two open questions raised by Herzog. Moreover, we show that the question whether $ M$ has Stanley depth at least $ r$ can be reduced to the question whether a certain combinatorially defined polytope $ \mathscr {P}$ contains a $ \mathbb{Z}^n$-lattice point.

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Additional Information

Bogdan Ichim
Affiliation: Simion Stoilow Institute of Mathematics of the Romanian Academy, Research Unit 5, C.P. 1-764, 014700 Bucharest, Romania

Lukas Katthän
Affiliation: FB Mathematik/Informatik, Universität Osnabrück, 49069 Osnabrück, Germany
Address at time of publication: Institut für Mathematik, Goethe-Universität Frankfurt, Robert-Mayer-Str. 10, 60325 Frankfurt am Main, Germany

Julio José Moyano-Fernández
Affiliation: Departamento de Matemáticas $&$ Institut Universitari de Matemàtiques i Aplicacions de Castelló, Universitat Jaume I, Campus de Riu Sec, 12071 Castellón de la Plana, Spain

Keywords: Graded modules, Hilbert depth, Stanley depth, Stanley decomposition.
Received by editor(s): April 2, 2015
Received by editor(s) in revised form: July 16, 2015
Published electronically: April 13, 2016
Additional Notes: The first author was partially supported by the project PN-II-RU-TE-2012-3-0161, granted by the Romanian National Authority for Scientific Research, CNCS – UEFISCDI
The second author was partially supported by the German Research Council DFG-GRK 1916
The third author was partially supported by the Spanish Government, Ministerio de Economía y Competitividad (MINECO), grants MTM2012-36917-C03-03 and MTM2015-65764-C3-2-P, as well as by Universitat Jaume I, grant P1-1B2015-02.
Article copyright: © Copyright 2016 American Mathematical Society