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An algorithm for construction of commuting ordinary differential operators by geometric data

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Abstract

We give a description and program realisation of an algorithm which produces explicit examples of commuting ordinary differential operators with rational coefficients by geometric data consisting of a rational curve and a torsion free sheaf with some trivialisation.

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References

  1. A. Beauville, M. S. Narasimhan, and S. Ramanan, “Spectral curves and the generalised theta divisor,” Reine Angew. Math. 398, 169–179 (1989).

    MathSciNet  MATH  Google Scholar 

  2. L. Bodnarchuk, I. Burban, Yu. Drozd, and G.-M. Greuel, “Vector bundles and torsion free sheaves on degenerations of elliptic curves,” in Global Aspects of Complex Geometry (Springer, Berlin, Heidelberg, 2006), pp. 83–128.

    Chapter  Google Scholar 

  3. I. Burban and Yu. Drozd, “Maximal Cohen–Macaulay modules over non-isolated surface singularities,” Memoirs AMS (2017, in press).

    Google Scholar 

  4. I. Burban and A. Zheglov, “Fourier–Mukai transform on Weierstrass cubics and commuting differential operators,” Oberwolfach Preprint No. 3 (Math. Forschungsinst. Oberwolfach, Oberwolfach, Germany, 2016), pp. 1–32.

    Google Scholar 

  5. J. L. Burchnall and T.W. Chaundy, “Commutative ordinary differential operators,” Proc. LondonMath. Soc., Ser. 2 21, 420–440 (1923); Proc. R. Soc. London, Ser. A 118, 557–583 (1928).

    Article  MATH  Google Scholar 

  6. V. N. Davletshina, “On self-adjoint commuting ordinary differential operators of rank two,” Siber. Electron. Math. Rep. 10, 109–112 (2013).

    MathSciNet  MATH  Google Scholar 

  7. P. G. Grinevich, “Rational solutions for the equation of commutation of differential operators,” Functional Anal. Appl. 16, 15–19 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  8. F. A. Grünbaum, “Commuting pairs of linear ordinary differential operators of orders four and six,” Physica D 31, 424–433 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Dixmier, “Sur les algèbres deWeyl,” Bull. Soc.Math. France 96, 209–242 (1968).

    Article  MathSciNet  MATH  Google Scholar 

  10. V. Drinfeld, “Commutative subrings of certain noncommutative rings,” Func. Anal. Appl. 11, 11–14 (1977).

    MathSciNet  Google Scholar 

  11. I.M. Krichever, “Integration of nonlinear equations by the methods of algebraic geometry,” Russ. Math. Surv. 32, 183–208 (1977).

    Article  MATH  Google Scholar 

  12. I. M. Krichever, “Commutative rings of ordinary linear differential operators,” Func. Anal. Appl. 12 (3), 20–31 (1978).

    Article  MathSciNet  MATH  Google Scholar 

  13. I. M. Krichever and S. P. Novikov, “Holomorphic bundles over algebraic curves and nonlinear equations,” Russ. Math. Surv. 35 (6), 47–68 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  14. H. Kurke, D. Osipov, and A. Zheglov, “Commuting differential operators and higher-dimensional algebraic varieties,” Sel. Math., New Ser. 20, 1159–1195 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  15. Y. Manin, “Algebraic aspects of nonlinear differential equations,” Itogi Nauki Tekh., Ser. Sovrem. Probl.Mat. 11, 5–152 (1978).

    MathSciNet  Google Scholar 

  16. A. E. Mironov, “Self-adjoint commuting ordinary differential operators,” Invent.Math. 197, 417–431 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  17. O. I. Mokhov, “Commuting differential operators of rank 3 and nonlinear equations,” Math. USSR Izv. 35, 629–655 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  18. O. I. Mokhov, “Commuting ordinary differential operators of arbitrary genus and arbitrary rank with polynomial coefficients,” Am.Math. Soc. Transl. 234, 323–336 (2014).

    MathSciNet  MATH  Google Scholar 

  19. M. Mulase, “Category of vector bundles on algebraic curves and infinite-dimensional Grassmannians,” Int. J.Math. 1, 293–342 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Mulase, “Algebraic theory of the KP equations,” in Perspectives in Mathematical Physics, Ed. by R. Penner and S. Yau (1994), pp. 151–218.

    Google Scholar 

  21. D. Mumford, “An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg de Vries equation and related nonlinear equation,” in Proceedings of the International Symposium on Algebra and Geometry, Kyoto, 1977 (Kinokuniya, 1978), pp. 115–153.

    Google Scholar 

  22. V. Oganesyan, “Commuting differential operators of rank 2 with polynomial coefficients,” Func. Anal. Appl. 50, 54–61 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  23. F. J. Plaza Martin, “Arithmetic infinite Grassmannians and the induced central extensions,” Collect. Math. 61, 107–129 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  24. E. Previato and G. Wilson, “Differential operators and rank 2 bundles over elliptic curves,” Comp.Math. 81, 107–119 (1992).

    MathSciNet  MATH  Google Scholar 

  25. I. Quandt, “On a relative version of the Krichever correspondence,” Bayreuther Math. Schrift. 52, 1–74 (1997).

    MathSciNet  MATH  Google Scholar 

  26. G. Wilson, “Algebraic curves and soliton equations,” J. Reine Angew.Math. 442, 177–204 (1993).

    MathSciNet  Google Scholar 

  27. A. E. Mironov and A. B. Zheglov, “Commuting ordinary differential operators with polynomial coefficients and automorphisms of the firstWeyl algebra,” Int. Math. Res. Not. 10, 2974–2993 (2016).

    Article  Google Scholar 

  28. A. B. Zheglov, A. E. Mironov, and B. T. Saparbaeva, “Commuting differential operators of Krichever–Novikov with polynomial coefficients,” Sib. Math. J. 57, 1048–1053 (2016).

    Article  MathSciNet  Google Scholar 

  29. A. B. Zheglov, “On rings of commuting differential operators,” SPb. Math. J. 25, 775–814 (2014).

    MathSciNet  MATH  Google Scholar 

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Pogorelov, D.A., Zheglov, A.B. An algorithm for construction of commuting ordinary differential operators by geometric data. Lobachevskii J Math 38, 1075–1092 (2017). https://doi.org/10.1134/S1995080217060117

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