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Transactions of the American Mathematical Society

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Systems of $ n$ partial differential equations in $ n$ unknown functions: the conjecture of M. Janet

Author: Joseph Johnson
Journal: Trans. Amer. Math. Soc. 242 (1978), 329-334
MSC: Primary 12H05; Secondary 58H05
MathSciNet review: 0491637
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Abstract: It was conjectured by Janet that an analytic solution to a system of n ``independent'' analytic differential equations in n unknown functions if not isolated must depend on at least one unknown function of $ m - 1$ variables plus possibly other functions of fewer than m variables. Here m is the dimension of the complex domain on which the equations and the solution are given. An algebraic generalization of the linear form of the conjecture is proven. Also the result is extended to give a nonlinear version.

References [Enhancements On Off] (What's this?)

  • [1] Maurice Janet, Sur les systemes aux dérivées partielles comprenant autant d'équations qui de fonctions inconnues, C. R. Acad. Sci. Paris 172 (1921), 1637-1639.
  • [2] K. R. Goodearl, Global dimension of differential operator rings. II, Trans. Amer. Math. Soc. 209 (1975), 65-85. MR 0382359 (52:3244)
  • [3] Joseph Johnson, Differential dimension polynomials and a fundamental theorem on differential modules, Amer. J. Math. 91 (1969), 239-248. MR 0238822 (39:186)
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  • [7] Ellis Kolchin, Some problems in differential algebra, Proc. Internat. Congr. Math., (Moscow, 1966), Izdat ``Mir'' Moscow, 1968, pp. 269-276. MR 0237476 (38:5758)

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Keywords: Differential algebra, differential ring, differential module, Kähler differentials
Article copyright: © Copyright 1978 American Mathematical Society

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