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Global properties of Kovalevskaya exponents. (English) Zbl 1398.37051

Summary: This paper contains a collection of properties of Kovalevskaya exponents which are eigenvalues of a linearization matrix of weighted homogeneous nonlinear systems along certain straight-line particular solutions. Relations in the form of linear combinations of Kovalevskaya exponents with nonnegative integers related to the presence of first integrals of the weighted homogeneous nonlinear systems have been known for a long time. As a new result other nonlinear relations between Kovalevskaya exponents calculated on all straight-line particular solutions are presented. They were obtained by an application of the Euler-Jacobi-Kronecker formula specified to an appropriate \(n\)-form in a certain weighted homogeneous projective space.

MSC:

37J30 Obstructions to integrability for finite-dimensional Hamiltonian and Lagrangian systems (nonintegrability criteria)
34M45 Ordinary differential equations on complex manifolds
32A27 Residues for several complex variables
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[1] Aizenberg, I.A. and Yuzhakov, A.P., Integral Representations and Residues in Multidimensional Complex Analysis, Transl. Math. Monogr., vol. 58, Providence,R.I.: AMS, 1983. · Zbl 0537.32002
[2] Borisov, A.V.; Dudoladov, S. L., Kovalevskaya exponents and Poisson structures, Regul. Chaotic Dyn., 4, 13-20, (1999) · Zbl 1012.37032 · doi:10.1070/rd1999v004n03ABEH000111
[3] Borisov, A.V.; Tsygvintsev, A.V., Kowalewski exponents and integrable systems of classical dynamics: 1, 2, Regul. Chaotic Dyn., 1, 15-37, (1996) · Zbl 1001.70506
[4] Cattani, E.; Dickenstein, A.; Sturmfels, B., Computing multidimensional residues, Algorithms in Algebraic Geometry and Applications (Santander, 1994), 143, 135-164, (1996) · Zbl 0882.13020 · doi:10.1007/978-3-0348-9104-2_8
[5] Chiba, H., Kovalevskaya exponents and the space of initial conditions of a quasi-homogeneous vector field, J. Differential Equations, 259, 7681-7716, (2015) · Zbl 1329.35012 · doi:10.1016/j.jde.2015.08.035
[6] Emel’yanov, K.V.; Tsygvintsev, A.V., Kovalevskaya exponents of systems with exponential interaction, Sb. Math., 191, 1459-1469, (2000) · Zbl 1156.37316 · doi:10.1070/SM2000v191n10ABEH000514
[7] Goriely, A., A brief history of Kovalevskaya exponents and modern developments, Regul. Chaotic Dyn., 5, 3-15, (2000) · Zbl 0947.37031 · doi:10.1070/rd2000v005n01ABEH000120
[8] Griffiths, P. and Harris, J., Principles of Algebraic Geometry, New York: Wiley, 1978. · Zbl 0408.14001
[9] Griffiths, P.A., Variations on a theorem of Abel, Invent. Math., 35, 321-390, (1976) · Zbl 0339.14003 · doi:10.1007/BF01390145
[10] Halphen, G.-H., Sur un système d’equations différentielles, C. R. Acad. Sci. Paris, 92, 1101-1103, (1881) · JFM 13.0289.02
[11] Halphen, G.-H., Sur certains systèmes d’équations différentielles, C. R. Acad. Sci. Paris, 92, 1404-1406, (1881) · JFM 13.0290.02
[12] Iano-Fletcher, A. R., Working with weighted complete intersections, Explicit Birational Geometry of 3-Folds, 281, 101-173, (2000) · Zbl 0960.14027 · doi:10.1017/CBO9780511758942.005
[13] Khimshiashvili, G., Multidimensional residues and polynomial equations, J. Math. Sci., 132, 757-804, (2006) · Zbl 1090.14017 · doi:10.1007/s10958-006-0021-1
[14] Kozlov, V.V., Tensor invariants of quasihomogeneous systems of differential equations, and the asymptotic Kovalevskaya-Lyapunov method, Math. Notes, 51, 138-142, (1992) · Zbl 0819.34004 · doi:10.1007/BF02102118
[15] Kozlov, V.V.; Treshchëv, D.V., Kovalevskaya numbers of generalized Toda chains, Math. Notes, 46, 840-848, (1989) · Zbl 0702.58066 · doi:10.1007/BF01139615
[16] Kummer, M.; Churchill, R. C.; Rod, D. L.; J, A. (ed.); H, ü (ed.), On kowaleskaya exponents, 71-76, (1991), Reading,Pa.
[17] Lochak, P., Une propriété des exposants de kowalevska des systèmes hamiltoniens: critère de Painlevé, C. R. Acad. Sci. Paris. Sér. 1 Math., 300, 369-372, (1985) · Zbl 0592.58017
[18] Lochak, P., Pairing of the kowalevska exponents in Hamiltonian systems, Phys. Lett. A, 108, 188-190, (1985) · doi:10.1016/0375-9601(85)90288-9
[19] Nowicki, A., Polynomial derivations and their rings of constants, 176, (1994) · Zbl 1236.13023
[20] Przybylska, M., Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom, Regul. Chaotic Dyn., 14, 263-311, (2009) · Zbl 1229.37059 · doi:10.1134/S1560354709020063
[21] Przybylska, M., Darboux points and integrability of homogeneous Hamiltonian systems with three and more degrees of freedom: nongeneric cases, Regul. Chaotic Dyn., 14, 349-388, (2009) · Zbl 1229.37060 · doi:10.1134/S1560354709030022
[22] Sadètov, S.T., Resonances on the Kovalevskaya exponent, Math. Notes, 54, 1081-1082, (1993) · Zbl 0820.58048 · doi:10.1007/BF01210428
[23] Sadètov, S.T., Resonances on the Kovalevskaya exponents and their memory on some tensor conservation laws, Vestn. Mosk. Univ. Ser. 1. Mat. Mekh., 1, 82-87, (1994) · Zbl 0887.70003
[24] Tsikh, A. K., Multidimensional Residues and Their Applications, Trans. Math. Monogr., vol. 103, Providence,R.I.: AMS, 1992. · Zbl 0758.32001
[25] Yoshida, H., Necessary condition for the existence of algebraic first integrals: 1. kowalevski’s exponents, Celest. Mech. Dyn. Astron., 31, 363-379, (1983) · Zbl 0556.70014 · doi:10.1007/BF01230292
[26] Yoshida, H., Necessary condition for the existence of algebraic first integrals: 2. condition for algebraic integrability, Celest. Mech. Dyn. Astron., 31, 381-399, (1983) · Zbl 0556.70015 · doi:10.1007/BF01230293
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