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Global Lie–Tresse theorem

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Abstract

We prove a global algebraic version of the Lie–Tresse theorem which states that the algebra of differential invariants of an algebraic pseudogroup action on a differential equation is generated by a finite number of rational-polynomial differential invariants and invariant derivations.

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Notes

  1. In the local case one restricts to a neighborhood \(\hat{U}=\pi _{\infty ,0}^{-1}(U)\subset J^\infty \) for an open set \(U\subset M\), while in the microlocal case \(\hat{U}=\pi _{\infty ,k}^{-1}(U)\subset J^\infty \) for some open set \(U\subset J^k\).

  2. Ultimately, when \(l\rightarrow \infty \), we get functions that are smooth in the base variables and rational on the fibers. The theory applies to this class as well (and is simpler).

  3. The equivalence problems of geometric structures on a manifold can be set up to be a partial case of our general setting for equivalence of submanifolds.

  4. When we turn to the algebraic situation, then “open” becomes “Zariski open.”

  5. In several sources, e.g., [38], \(\varphi \) is called characteristic, but in our case this would lead to a confusion with characteristic covectors, and we use the terminology adapted in [26].

  6. To keep the notations simple, we denote the required subset in the set of regular points by the same symbol \({\mathcal {E}}_l''\subset {\mathcal {E}}_l'\) (we also abuse notations by redefining l several times).

  7. Though the Lie–Tresse–Kumpera pseudogroup is not transitive (and so formally our theorems are not applicable), this is the case where our theory still works.

  8. Over characteristic 0 this theorem states that a finite extension E of a field K is generated by one element \(E=K(\alpha )\), see, e.g., [31, V. §4: Theorem 4.6].

  9. Here by \(\dim \mathcal {A}\) we understand the maximal number of functionally independent differential invariants (recall that any function of differential invariants is a differential invariant itself, so the dimension of \(\mathcal {A}\) as a vector space is either 1 or \(\infty \)).

  10. The are two obvious choices for the invariant differentiations: derivatives \(\mathfrak {D}({\mathcal {E}},\tau )\) and derivations \(\mathfrak {Der}({\mathcal {E}},\tau )\). We choose the latter, because the primary goal is to generate \({\mathcal {A}}\).

  11. Here D means a module over the algebra \(\mathfrak {Diff}({\mathcal {E}},\tau )\). Notice that the algebra \(\mathfrak {Diff}({\mathcal {E}},\tau )\) is generated by \({\mathcal {A}}\) and \(\nabla _j\), and so is finitely generated by \(I_i\) and \(\nabla _j\).

  12. Notice that the Hilbert and Poincaré functions are related by \(H^{\mathcal {E}}_G(k)=\frac{1}{k!}\left. \frac{{\text {d}}^k}{{\text {d}}z^k}\right| _{z=0}P^{\mathcal {E}}_G(z)\).

  13. Some people interpret them as the microlocal differential invariants, but we believe that the moduli are meant to be the actual global invariants.

  14. This makes the group action intransitive, but it is irrelevant for this example. Simply add the sixth generator \(\partial _u\) to \({\mathfrak {g}}\), or observe that all our constructions work without this.

  15. The condition of “no resonances” can be omitted since the functions \(I_2,I_4,\dots \) below are invariants anyway, but this condition relates them to the Birkhoff normal form and shows that no more independent invariants exist.

References

  1. Arnold, V.I.: Mathematical problems in classical physics, Trends and perspectives in applied mathematics. Appl. Math. Sci. 100, 1–20, Springer-Verlag (1994); also: Arnold’s problems, Fazis, Moscow (2000); Springer, Berlin (2004)

  2. Auslander, M., Buchsbaum, D.A.: Codimension and multiplicity. Ann. Math. 68(3), 625–657 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alekseevskij, D., Lychagin, V., Vinogradov, A.: Basic Ideas and Concepts of Differential Geometry, Encyclopedia of Mathematical Sciences, Geometry 1, vol. 28. Springer, Berlin (1991)

    Google Scholar 

  4. Albert, C., Molino, P.: Pseudogroupes de Lie transitifs. II, Travaux en Cours 19. Hermann, Paris (1987)

    MATH  Google Scholar 

  5. Anderson, I.M., Kruglikov, B.: Rank 2 distributions of Monge equations: symmetries, equivalences, extensions. Adv. Math. 228(3), 1435–1465 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Birkhoff, G.D.: Dynamical Systems. Colloquium Publication IX. American Mathematical Society, Providence (1927)

    Google Scholar 

  7. Bibikov, P.V., Lychagin, V.V.: Projective classification of binary and ternary forms. J. Geom. Phys. 61(10), 1914–1927 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cartan, E.: Sur la structure des groupes infinis de transformations. Ann. Sci. E’cole Norm. Sup. (3) 21, 153–206 (1904); 22, 219–308 (1905)

  9. Cartan, E.: Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre. Ann. Sci. École Norm. Sup. (3) 27, 109–192 (1910)

    MathSciNet  MATH  Google Scholar 

  10. Chevalley, C.: A new kind of relationship between matrices. Am. J. Math. 65, 521–531 (1943)

    Article  MathSciNet  MATH  Google Scholar 

  11. Doubrov, B., Kruglikov, B.: On the models of submaximal symmetric rank 2 distributions in 5D. Differ. Geom. Appl. 35, 314–322 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dubois, D.W.: A Nullstellensatz for ordered fields. Ark. Mat. 8, 111–114 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  13. Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry. Springer, Berlin (1995)

    MATH  Google Scholar 

  14. Goldberg, V.V., Lychagin, V.: Geodesic webs on a two-dimensional manifold and Euler equations. Acta Appl. Math. 109(1), 5–17 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goldschmidt, H.: Integrability criteria for systems of nonlinear partial differential equations. J. Differ. Geom. 1(3), 269–307 (1967)

    MathSciNet  MATH  Google Scholar 

  16. Greb, D., Miebach, C.: Invariant meromorphic functions on Stein spaces. Ann. Inst. Fourier (Grenoble) 62(5), 1983–2011 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Guillemin, V., Sternberg, S.: An algebraic model of transitive differential geometry. Bull. Am. Math. Soc. 70, 16–47 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  18. Guillemin, V., Sternberg, S.: Deformation theory of pseudogroup structures. Mem. Am. Math. Soc. 64, 1–80 (1966)

    MathSciNet  MATH  Google Scholar 

  19. Halpen, M.: Sur les invariants differentiels. Thèse, Paris (1878)

  20. Hubert, E., Kogan, I.: Smooth and algebraic invariants of a group action: local and global constructions. Found. Comput. Math. 7(4), 455–493 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. Klein, F.: Vorlesungen über höhere Geometrie, Die Grundlehren der mathematischen Wissenschaften 22. Springer, Berlin (1926)

    Book  Google Scholar 

  22. Kruglikov, B.S., Lychagin, V.V.: Mayer brackets and solvability of PDEs—II. Trans. Am. Math. Soc. 358(3), 1077–1103 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kruglikov, B.S., Lychagin, V.V.: Invariants of pseudogroup actions: homological methods and finiteness theorem. Int. J. Geom. Methods Mod. Phys. 3(5&6), 1131–1165 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kruglikov, B.S., Lychagin, V.V.: Spencer \(\delta \)-cohomology, restrictions, characteristics and involutive symbolic PDEs. Acta Appl. Math. 95, 31–50 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kruglikov, B.S., Lychagin, V.V.: Geometry of differential equations. In: Krupka, D., Saunders, D. (eds.) Handbook of Global Analysis, pp. 725–772. Elsevier, Amsterdam (2008)

    Chapter  Google Scholar 

  26. Krasilschik, I.S., Lychagin, V.V., Vinogradov, A.M.: Geometry of Jet Spaces and Differential Equations. Gordon and Breach, New York (1986)

    Google Scholar 

  27. Kruglikov, B.: Point classification of second order ODEs: Tresse classification revisited and beyond (with an appendix by Kruglikov and V.Lychagin). Abel Symposium 5, Differential equations: geometry, symmetries and integrability, pp. 199–221, Springer, Berlin (2009)

  28. Krupka, D., Janyška, L.: Lectures on Differential Invariants. Folia University J.E. Purkyne, Brno (1990)

    MATH  Google Scholar 

  29. Kumpera, A.: Invariants differentiels d’un pseudogroupe de Lie. I–II. J. Differ. Geom. 10(2), 289–345; (3), 347–416 (1975)

  30. Kuranishi, M.: On E. Cartan’s prolongation theorem of exterior differential systems. Am. J. Math. 79, 1–47 (1957)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lang, S.: Algebra, 3rd edn. Addison-Wesley Pub. Co., Reading (1993)

    MATH  Google Scholar 

  32. Lie, S.: Theorie der Transformationsgruppen (Zweiter Abschnitt, unter Mitwirkung von Prof.Dr.Friederich Engel). Teubner, Leipzig (1890)

    MATH  Google Scholar 

  33. Lie, S.: Ueber Differentialinvarianten. Math. Ann. 24(4), 537–578 (1884); also: Sophus Lie’s 1884 differential invariant paper, Engl. translation by M. Ackerman, comments by R. Hermann, Math. Sci. Press, Brookline, MA (1976)

  34. Lychagin, V.: Homogeneous geometric structures and homogeneous differential equations. A.M.S. Transl. Ser., The interplay between differential geometry and differential equations, ser. 2, 167, 143–164 (1995)

  35. Malgrange, B.: Systèmes différentiels involutifs, Panoramas et Synthèses 19, Soc. Math. de France (2005)

  36. Mansfield, E.: A Practical Guide to the Invariant Calculus, Cambridge Monographs on Applied and Computational Mathematics 26. Cambridge University Press, Cambridge (2010)

    Book  Google Scholar 

  37. Munoz, J., Muriel, F.J., Rodriguez, J.: On the finiteness of differential invariants. J. Math. Anal. Appl. 284(1), 266–282 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  38. Olver, P.: Applications of Lie groups to Differential Equations, Graduate Texts in Mathematics, vol. 107. Springer, New York (1986)

    Book  Google Scholar 

  39. Olver, P.: Classical Invariant Theory. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  40. Olver, P.: Differential invariant algebras. Comtemp. Math. 549, 95–121 (2011)

    MathSciNet  MATH  Google Scholar 

  41. Olver, P., Pohjanpelto, J.: Differential invariant algebras of Lie pseudo-groups. Adv. Math. 222(5), 1746–1792 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ovsiannikov, L.V.: Group analysis of differential equations, Russian: Nauka, Moscow (1978); Engl. trans.: Academic Press, New York (1982)

  43. Popov, V.L., Vinberg, E.B.: Invariant theory. In: Algebraic geometry. IV, Encyclopaedia of Mathematical Sciences 55 (translation from Russian edited by A. N. Parshin, I. R. Shafarevich), Springer-Verlag, Berlin (1994)

  44. Risler, J.-J.: Une caractérisation des idéaux des variétés algébriques réelles. C.R.Acad.Sci. Paris Sér. A-B 271, A1171–A1173 (1970)

    MathSciNet  MATH  Google Scholar 

  45. Ritt, J.F.: Differential Algebra, vol. 33. American Mathematical Society Colloquium Publications, New York (1950)

    MATH  Google Scholar 

  46. Rosenlicht, M.: Some basic theorems on algebraic groups. Am. J. Math. 78, 401–443 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  47. Rosenlicht, M.: A remark on quotient spaces. An. Acad. Brasil. Ci. 35, 487–489 (1963)

    MathSciNet  MATH  Google Scholar 

  48. Robart, T., Kamran, N.: Sur la théorie locale des pseudogroupes de transformations continus infinis. I. Math. Ann. 308(4), 593–613 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  49. Santos, W.F., Rittatore, A.: Actions and Invariants of Algebraic Groups. Chapman & Hall/CRC Press, Boca Raton (2005)

    Book  MATH  Google Scholar 

  50. Sarkisyan, R.A.: Rationality of the Poincaré series in Arnold’s local problems of analysis. Izvestiya RAN Ser. Mat. (Izv.Math.) 74(2), 411–438 (2010)

    MathSciNet  MATH  Google Scholar 

  51. Sarkisyan, R.A., Shandra, I.G.: Regularity and Tresse’s theorem for geometric structures. Izv. Ross. Akad. Nauk Ser. Mat. 72(2), 151–192; Engl.transl. in. Izv. Math. 72(2), 345–382 (2008)

  52. Seiler, W.: Involution. The Formal Theory of Differential Equations and Its Applications in Computer Algebra, Algorithms and Computation in Mathematics, vol. 24. Springer, Berlin (2010)

    MATH  Google Scholar 

  53. Singer, I.M., Sternberg, S.: On the infinite groups of Lie and Cartan. J. d’Analyse Math. 15, 1–114 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  54. Spencer, D.: Deformation of structures on manifolds defined by transitive pseudogroups. Ann. Math. 76(2), 306–445 (1962)

    Article  MATH  Google Scholar 

  55. Spencer, D.C.: Overdetermined systems of linear partial differential equations. Bull. Am. Math. Soc. 75, 179–239 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  56. Study, E.: Die Elemente zweiter Ordnung in der ebenen projektiven Geometrie. Leipziger Berichte 53, 338–403 (1901)

    MATH  Google Scholar 

  57. Thomas, T.V.: The Differential Invariants of Generalized Spaces. The University Press, Cambridge (1934)

    MATH  Google Scholar 

  58. Tresse, A.: Sur les invariants différentiels des groupes continus de transformations. Acta Math. 18, 1–88 (1894)

    Article  MathSciNet  MATH  Google Scholar 

  59. Tresse, A.: Détermination des invariants ponctuels de l’équation différentielle ordinaire de second ordre \(y^{\prime \prime }=\omega (x, y, y^{\prime })\). Mémoire couronné par l’Académie Jablonowski. S. Hirkel, Leipzig (1896)

    MATH  Google Scholar 

  60. Valiquette, F.: Solving local equivalence problems with the equivariant moving frame method. SIGMA 9, 029 (2013)

    MathSciNet  MATH  Google Scholar 

  61. Veblen, O.: Differential Invariants and Geometry. Atti del Congresso Internazionale dei Matematici. Bologna, Nicola Zanichelli, pp. 181–190 (1928). Available at: http://www.mathunion.org/ICM/#1928

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  1. Institute of Mathematics and Statistics, University of Tromsø, 90-37, Tromsø, Norway

    Boris Kruglikov & Valentin Lychagin

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Kruglikov, B., Lychagin, V. Global Lie–Tresse theorem. Sel. Math. New Ser. 22, 1357–1411 (2016). https://doi.org/10.1007/s00029-015-0220-z

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