Ir al contenido

Documat

Telescopers for differential forms with one parameter

  • Shaoshi Chen [1] ; Ruyong Feng [1] ; Ziming Li [1] ; Michael F Singer [2] ; Stephen M. Watt [3]
    1. [1] University of Chinese Academy of Sciences

      University of Chinese Academy of Sciences

      China

    2. [2] North Carolina State University

      North Carolina State University

      Township of Raleigh, Estados Unidos

    3. [3] University of Waterloo

      University of Waterloo

      Canadá

    Mostrar afiliaciones +
  • Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 3, 2024, págs. 1-23
  • Idioma: inglés
  • DOI: 10.1007/s00029-024-00926-6
  • Enlaces
  • Resumen

    • Telescopers for a function are linear differential (resp. difference) operators annihilating the definite integral (resp. definite sum) of this function. They play a key role in Wilf–Zeilberger theory and algorithms for computing them have been extensively studied in the past 30 years. In this paper, we introduce the notion of telescopers for differential forms with D-finite function coefficients. These telescopers appear in several areas of mathematics, for instance parametrized differential Galois theory and mirror symmetry. We give a sufficient and necessary condition for the existence of telescopers for a differential form and describe a method to compute them if they exist. Algorithms for verifying this condition are also given.

  • Referencias bibliográficas
    • Abramov, S.A.: When does Zeilberger’s algorithm succeed? Adv. Appl. Math. 30(3), 424–441 (2003).
    • Abramov, S.A., Le, H.Q.: A criterion for the applicability of Zeilberger’s algorithm to rational functions. Discrete Math. 259(1–3), 1–17...
    • Almkvist, G., Zeilberger, D.: The method of differentiating under the integral sign. J. Symb. Comput. 10(6), 571–591 (1990).
    • Beke, E.: Die Irreducibilität der homogenen linearen Differentialgleichungen. Math. Ann. 45(2), 278–294 (1894).
    • Bernšte˘ın, I.N.: Modules over a ring of differential operators. An investigation of the fundamental solutions of equations with constant...
    • Björk, J.-E.: Rings of differential operators. In: North-Holland Mathematical Library, vol. 21. North-Holland Publishing Co., Amsterdam (1979).
    • Bostan, A., Chen, S., Chyzak, F., Li, Z.: Complexity of creative telescoping for bivariate rational functions. In: ISSAC 2010—Proceedings...
    • Bostan, A., Chen, S., Chyzak, F., Li, Z., Xin, G.: Hermite reduction and creative telescoping for hyperexponential functions. In: ISSAC 2013—Proceedings...
    • Bostan, A., Chyzak, F., Lairez, P., Salvy, B.: Generalized Hermite reduction, creative telescoping and definite integration of D-finite functions....
    • Bostan, A., Lairez, P., Salvy, B.: Creative telescoping for rational functions using the Griffiths–Dwork method. In: ISSAC 2013—Proceedings...
    • Chen, S., Chyzak, F., Feng, R., Fu, G., Li, Z.: On the existence of telescopers for mixed hypergeometric terms. J. Symb. Comput. 68(part 1),...
    • Chen, S., Feng, R., Li, Z., Singer, M.F.: Parallel telescoping and parameterized Picard–Vessiot theory. In: ISSAC 2014—Proceedings of the...
    • Chen, S., Feng, R., Ma, P., Singer, M.F.: Separability problems in creative telescoping. In: ISSAC 2021—Proceedings of the 2021 International...
    • Chen, S., Hou, Q.-H., Labahn, G., Wang, R.-H.: Existence problem of telescopers: beyond the bivariate case. In: ISSAC 2016—Proceedings of...
    • Chen, S., Kauers, M.: Some open problems related to creative telescoping. J. Syst. Sci. Complex. 30(1), 154–172 (2017).
    • Chen, S., Kauers, M., Koutschan, C.: Reduction-based creative telescoping for algebraic functions. In: ISSAC 2016—Proceedings of the 2016...
    • Chen, S., Kauers, M., Li, Z., Zhang, Y.: Apparent singularities of D-finite systems. J. Symb. Comput. 95, 217–237 (2019).
    • Chen, S., Kauers, M., Singer, M.F.: Desingularization of ore operators. J. Symb. Comput. 74(C), 617–626 (2016).
    • Chen, S., Koutschan, C.: Proof of the Wilf–Zeilberger conjecture for mixed hypergeometric terms. J. Symb. Comput. 93, 133–147 (2019).
    • Chen, S., van Hoeij, M., Kauers, M., Koutschan, C.: Reduction-based creative telescoping for fuchsian D-finite functions. J. Symb. Comput....
    • Chen, W.Y.C., Hou, Q.-H., Mu, Y.-P.: Applicability of the q-analogue of Zeilberger’s algorithm. J. Symb. Comput. 39(2), 155–170 (2005).
    • Euler, L.: Specimen de constructione aequationum differentialium sine indeterminatarum separatione. Commentarii academiae scientiarum Petropolitanae...
    • Kashiwara, M.: On the holonomic systems of linear differential equations II. Invent. Math. 49(2), 121–135 (1978).
    • Kolchin, E.R.: Differential algebra and algebraic groups. In: Pure and Applied Mathematics, vol. 54. Academic Press, New York (1973).
    • Koutschan, C.: Advanced applications of the holonomic systems approach. Ph.D thesis, Research Institute for Symbolic Computation (RISC), Johannes...
    • Koutschan, C.: Creative telescoping for holonomic functions. In: Computer Algebra in Quantum Field Theory, Texts Monogram Symbol Computer,...
    • Lairez, P.: Computing periods of rational integrals. Math. Comput. 85(300), 1719–1752 (2016).
    • Lairez, P., Vanhove, P.: Algorithms for minimal Picard–Fuchs operators of Feynman integrals. Lett. Math. Phys. 113(2), 37 (2023).
    • Lang, S.: Algebra, Volume 211 of Graduate Texts in Mathematics, 3rd edn. Springer, New York (2002).
    • Li, S., Lian, B.H., Yau, S.-T.: Picard–Fuchs equations for relative periods and Abel–Jacobi map for Calabi–Yau hypersurfaces. Am. J. Math....
    • Li, Z., Wang, H.: A criterion for the similarity of length-two elements in a noncommutative PID. J. Syst. Sci. Complex. 24(3), 580–592 (2011).
    • Lipshitz, L.: The diagonal of a D-finite power series is D-finite. J. Algebra 113(2), 373–378 (1988).
    • Morrison, D.R., Walcher, J.: D-branes and normal functions. Adv. Theor. Math. Phys. 13(2), 553–598 (2009).
    • Müller-Stach, S., Weinzierl, S., Zayadeh, R.: Picard–Fuchs equations for Feynman integrals. Commun. Math. Phys. 326(1), 237–249 (2014).
    • Ore, O.: Theory of non-commutative polynomials. Ann. Math. (2) 34(3), 480–508 (1933).
    • Petkovšek, M., Wilf, H.S., Zeilberger, D.: A = B. AK Peters Ltd., Wellesley, MA: With a foreword by Donald E. Knuth, With a Separately...
    • Schwarz, F.: Loewy decomposition of linear differential equations. In: Texts and Monographs in Symbolic Computation. Springer, New York (2012).
    • Singer, M.F.: Introduction to the Galois theory of linear differential equations. In: Algebraic theory of differential equations, volume 357...
    • van der Poorten, A.: A proof that Euler missed... Apéry’s proof of the irrationality of ζ(3). Math. Intell. 1(4), 195–203 (1978/1979).
    • van der Put, M., Singer, M.F.: Galois theory of linear differential equations. In: Grundlehren der Mathematischen Wissenschaften [Fundamental...
    • van Hoeij, M.: Rational solutions of the mixed differential equation and its application to factorization of differential operators. In: Engeler,...
    • van Hoeij, M.: Factorization of differential operators with rational functions coefficients. J. Symb. Comput. 24(5), 537–561 (1997).
    • Weintraub, S.H.: Differential Forms: Theory and Practice. Elsevier/Academic Press, Amsterdam, 2nd edn (2014).
    • Wilf, H.S., Zeilberger, D.: Rational functions certify combinatorial identities. J. Am. Math. Soc. 3(1), 147–158 (1990).
    • Wilf, H.S., Zeilberger, D.: An algorithmic proof theory for hypergeometric (ordinary and “q”) multisum/integral identities. Invent. Math....
    • Wilf, H.S., Zeilberger, D.: Rational function certification of multisum/integral/“q” identities. Bull. Am. Math. Soc. (N.S.) 27(1), 148–153...
    • Zeilberger, D.: A holonomic systems approach to special functions identities. J. Comput. Appl. Math. 32(3), 321–368 (1990).
    • Zeilberger, D.: The method of creative telescoping. J. Symb. Comput. 11(3), 195–204 (1991).
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno