Abstract
We give an approximation algorithm of computing holonomic systems of linear differential equations for definite integrals of rational functions with parameters. We show that this algorithm gives a correct answer in finite steps, but we have no general stopping condition. We apply the approximation method to find differential equations for integrals associated to smooth Fano polytopes. These are interesting in the study of K3 surfaces and the toric mirror symmetry. In this class of integrals, we can apply Stienstra’s rank formula to our algorithm, which gives a stopping condition of the approximation algorithm.
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References
Batyrev V.: Variations of the mixed Hodge structure of affine hypersurfaces in algebaic tori. Duke Math. J. 69(2), 349–409 (1993)
Briançon, J., Maisonobe, P.: Remarques sur l’idéal de Bernstein associé à des polynômes, Preprint no.650, Univ. Nice Sophia-Antipolis (2002)
Castro-Jiménez F.J., Ucha-Enriquez J.M.: Gröbner bases and logarithmic \({\mathcal{D}}\) -modules. J. Symbolic Comput. 41, 317–335 (2006)
Chyzak F.: An extension of Zeilberger’s fast algorithm to general holonomic functions. Discrete Math. 217, 115–134 (2000)
Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-2—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de
Ishige, T.: An Isomorphic correspondence between the Hilbert modular group and the unimodular group of a domain of Type IV, Technical Report, Chiba University (2008)
Ishige, T.: A Family of K3 Surfaces connected with the Hilbert Modular Group for \({\sqrt{2}}\) and the GKZ Hypergeometric Differential Equation, preprint (2010)
Kreuzer, M., Skarke, H.: Classification of Reflexive Polyhedra in Three Dimensions, Advances in Theoretical and Mathematical Physics, 847–864 (1998)
Kreuzer, M., Skarke, H.: Calabi Yau Data, http://hep.itp.tuwien.ac.at/~kreuzer/CY/
Levandovskyy V., Martin Morales J.: Computational D-module theory with SINGULAR, comparison with other systems and two new algorithms. ISSAC 2008, 173–180 (2008)
Nagano A.: Period differential equations for the families of K3 surfaces with two parameters derived from the reflexive polytopes. Kyushu J. Math. 66(1), 193–244 (2012)
Noro, M. et al.: Risa/Asir, http://www.math.kobe-u.ac.jp/Asir
Oaku T.: Algorithm for the b-function and D-modules associated with a polynomial. J. Pure Appl. Algebra. 117(118), 495–518 (1997)
Oaku T.: Algorithms for b-functions, restrictions, and algebraic local cohomology groups of D-modules. Adv. Appl. Math. 19, 61–105 (1997)
Oaku T., Takayama N.: An Algorithm for de Rham cohomology groups of the complement of an affine variety via D-module computation. J. Pure Appl. Algebra. 139, 201–233 (1999)
Øbro, M.: Classification of Smooth Fano Polytopes, PhD thesis, Aarhus University (2007)
Saito M.: Irreducible Quotients of A-Hypergeometric Systems. Compositio Mathematica 147, 613–632 (2011)
Saito M., Sturmfels B., Takayama N.: Groebner Deformations of Hypergeometric Differential Equations. Springer, Berlin (1999)
Stienstra J.: Resonant hypergeometric systems and mirror symmetry. Integrable Systems and Algebraic Geometry Proceedings of the Taniguchi Symposium 1997, 412–452 (1998)
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Nakayama, H., Takayama, N. Computing differential equations for integrals associated to smooth Fano polytope. Japan J. Indust. Appl. Math. 30, 307–319 (2013). https://doi.org/10.1007/s13160-013-0105-5
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DOI: https://doi.org/10.1007/s13160-013-0105-5