Computing periods of rational integrals

Lairez, Pierre

doi:10.1090/mcom/3054

Computing periods of rational integrals
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Math. Comp. 85 (2016), 1719-1752 Request permission

Abstract:

A period of a rational integral is the result of integrating, with respect to one or several variables, a rational function over a closed path. This work focuses particularly on periods depending on a parameter: in this case the period under consideration satisfies a linear differential equation, the Picard-Fuchs equation. I give a reduction algorithm that extends the Griffiths-Dwork reduction and apply it to the computation of Picard-Fuchs equations. The resulting algorithm is elementary and has been successfully applied to problems that were previously out of reach.

References

Gert Almkvist, The art of finding Calabi-Yau differential equations. Dedicated to the 90-th birthday of Lars Gärding, Gems in experimental mathematics, Contemp. Math., vol. 517, Amer. Math. Soc., Providence, RI, 2010, pp. 1–18. MR 2731057, DOI 10.1090/conm/517/10129
G. Almkvist, Christian van Enckevort, Duco van Straten, and Wadim Zudilin, Tables of Calabi–Yau equations. arXiv:math/0507430, 2010.
Moa Apagodu and Doron Zeilberger, Multi-variable Zeilberger and Almkvist-Zeilberger algorithms and the sharpening of Wilf-Zeilberger theory, Adv. in Appl. Math. 37 (2006), no. 2, 139–152. MR 2251432, DOI 10.1016/j.aam.2005.09.003
Victor Batyrev and Maximilian Kreuzer, Constructing new Calabi-Yau 3-folds and their mirrors via conifold transitions, Adv. Theor. Math. Phys. 14 (2010), no. 3, 879–898. MR 2801412
Victor V. Batyrev and Duco van Straten, Generalized hypergeometric functions and rational curves on Calabi-Yau complete intersections in toric varieties, Comm. Math. Phys. 168 (1995), no. 3, 493–533. MR 1328251
F. Beukers, Irrationality of $\pi ^{2}$, periods of an elliptic curve and $\Gamma _{1}(5)$, Diophantine approximations and transcendental numbers (Luminy, 1982), Progr. Math., vol. 31, Birkhäuser, Boston, Mass., 1983, pp. 47–66. MR 702189
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
Alin Bostan, Shaoshi Chen, Frédéric Chyzak, and Ziming Li, Complexity of creative telescoping for bivariate rational functions, ISSAC 2010—Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2010, pp. 203–210. MR 2920555, DOI 10.1145/1837934.1837975
Alin Bostan, Pierre Lairez, and Bruno Salvy, Creative telescoping for rational functions using the Griffiths-Dwork method, ISSAC 2013—Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2013, pp. 93–100. MR 3206345, DOI 10.1145/2465506.2465935
N. Bourbaki, Algèbres tensorielles, algèbres extérieures, algèbres symétriques, Algèbre, Éléments de mathématiques, Chap. III, Hermann, 1961.
Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, Algorithmic probability and combinatorics, Contemp. Math., vol. 520, Amer. Math. Soc., Providence, RI, 2010, pp. 1–39. MR 2681853, DOI 10.1090/conm/520/10252
Shaoshi Chen, Manuel Kauers, and Michael F. Singer, Telescopers for rational and algebraic functions via residues, ISSAC 2012—Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation, ACM, New York, 2012, pp. 130–137. MR 3206296, DOI 10.1145/2442829.2442851
Frédéric Chyzak, An extension of Zeilberger’s fast algorithm to general holonomic functions, Discrete Math. 217 (2000), no. 1-3, 115–134 (English, with English and French summaries). Formal power series and algebraic combinatorics (Vienna, 1997). MR 1766263, DOI 10.1016/S0012-365X(99)00259-9
F. Chyzak, The ABC of Creative Telescoping: Algorithms, Bounds, Complexity, Mémoire d’habilitation à diriger les recherches, 2014.
David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, Mathematical Surveys and Monographs, vol. 68, American Mathematical Society, Providence, RI, 1999. MR 1677117, DOI 10.1090/surv/068
David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811, DOI 10.1007/978-1-4757-6911-1
P. Deligne, Lettre à Bernard Malgrange, April 30, 1991.
Alexandru Dimca, On the de Rham cohomology of a hypersurface complement, Amer. J. Math. 113 (1991), no. 4, 763–771. MR 1118460, DOI 10.2307/2374846
Alexandru Dimca, On the Milnor fibrations of weighted homogeneous polynomials, Compositio Math. 76 (1990), no. 1-2, 19–47. Algebraic geometry (Berlin, 1988). MR 1078856
Alexandru Dimca, Singularities and topology of hypersurfaces, Universitext, Springer-Verlag, New York, 1992. MR 1194180, DOI 10.1007/978-1-4612-4404-2
Alexandru Dimca and Morihiko Saito, A generalization of Griffiths’s theorem on rational integrals, Duke Math. J. 135 (2006), no. 2, 303–326. MR 2267285, DOI 10.1215/S0012-7094-06-13523-8
Bernard Dwork, On the zeta function of a hypersurface, Inst. Hautes Études Sci. Publ. Math. 12 (1962), 5–68. MR 159823
Bernard Dwork, On the zeta function of a hypersurface. II, Ann. of Math. (2) 80 (1964), 227–299. MR 188215, DOI 10.2307/1970392
L. Euler, Specimen de constructione aequationum differentialium sine indeterminatarum separatione, Commentarii academiae scientiarum Petropolitanae 6 (1733) (Opera omnia, 1e série, t. XX), 168–174.
Mary Celine Fasenmyer, Some generalized hypergeometric polynomials, Bull. Amer. Math. Soc. 53 (1947), 806–812. MR 22276, DOI 10.1090/S0002-9904-1947-08893-5
Joachim von zur Gathen and Jürgen Gerhard, Modern computer algebra, Cambridge University Press, New York, 1999. MR 1689167
Phillip A. Griffiths, On the periods of certain rational integrals. I, II, Ann. of Math. (2) 90 (1969), 460-495; ibid. (2) 90 (1969), 496–541. MR 0260733, DOI 10.2307/1970746
A. Grothendieck, On the de Rham cohomology of algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 29 (1966), 95–103. MR 199194
Ch. Hermite, Sur l’intégration des fractions rationnelles, Ann. Sci. École Norm. Sup. (2) 1 (1872), 215–218 (French). MR 1508584
Mark van Hoeij, Factorization of differential operators with rational functions coefficients, J. Symbolic Comput. 24 (1997), no. 5, 537–561. MR 1484068, DOI 10.1006/jsco.1997.0151
Nicholas M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175–232. MR 291177
Manuel Kauers and Doron Zeilberger, The computational challenge of enumerating high-dimensional rook walks, Adv. in Appl. Math. 47 (2011), no. 4, 813–819. MR 2832378, DOI 10.1016/j.aam.2011.03.004
Christoph Koutschan, A fast approach to creative telescoping, Math. Comput. Sci. 4 (2010), no. 2-3, 259–266. MR 2775992, DOI 10.1007/s11786-010-0055-0
Bernard Malgrange, Lettre à Pierre Deligne, April 26, 1991.
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
P. Metelitsyn, How to compute the constant term of a power of a Laurent polynomial efficiently, CoRR (2012). arXiv: abs/1211.3959
P. Monsky, Finiteness of de Rham cohomology, Amer. J. Math. 94 (1972), 237–245. MR 301017, DOI 10.2307/2373603
David R. Morrison, Picard-Fuchs equations and mirror maps for hypersurfaces, Essays on mirror manifolds, Int. Press, Hong Kong, 1992, pp. 241–264. MR 1191426
David R. Morrison and Johannes Walcher, D-branes and normal functions, Adv. Theor. Math. Phys. 13 (2009), no. 2, 553–598. MR 2481273
Toshinori Oaku and Nobuki Takayama, An algorithm for de Rham cohomology groups of the complement of an affine variety via $D$-module computation, J. Pure Appl. Algebra 139 (1999), no. 1-3, 201–233. Effective methods in algebraic geometry (Saint-Malo, 1998). MR 1700544, DOI 10.1016/S0022-4049(99)00012-2
È. Picard, Sur les intégrales doubles de fonctions rationnelles dont tous les résidus sont nuls, Bulletin des sciences mathématiques, série 2 26 (1902).
É. Picard and G. Simart, Théorie des fonctions algébriques de deux variables indépendantes. Vol. I. Gauthier-Villars et fils, 1897.
É. Picard and G. Simart, Théorie des fonctions algébriques de deux variables indépendantes. Vol. II. Gauthier-Villars et fils, 1906.
D. van Straten, Calabi-Yau Operators Database, http://www.mathematik.uni-mainz.de/ CYequations/db/.
P. Verbaeten, The automatic construction of pure recurrence relations, SIGSAM Bull. 8.3 (August 1974), 96–98.
Herbert S. Wilf and Doron Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and “$q$”) multisum/integral identities, Invent. Math. 108 (1992), no. 3, 575–633. MR 1163239, DOI 10.1007/BF02100618
Doron Zeilberger, The method of creative telescoping, J. Symbolic Comput. 11 (1991), no. 3, 195–204. MR 1103727, DOI 10.1016/S0747-7171(08)80044-2