Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The Greenspan bound for the order of differential systems


Author: Richard M. Cohn
Journal: Proc. Amer. Math. Soc. 79 (1980), 523-526
MSC: Primary 12H05
DOI: https://doi.org/10.1090/S0002-9939-1980-0572294-0
MathSciNet review: 572294
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let S be a system of ordinary differential polynomials in indeterminates $ {y_1}, \ldots ,{y_n}$ and of order at most $ {r_i}$ in $ {y_i},1 \leqslant i \leqslant n$. It was shown by J. F. Ritt that if $ \mathfrak{M}$ is a component of S of differential dimension 0, then the order of $ \mathfrak{M}$ is at most $ {r_1} + \ldots + {r_n}$. B. Greenspan improved this bound in the case that every component of S has differential dimension 0. (His work was carried out for difference equations, but is easily transferred to the differential case.) It is shown that the Greenspan bound is valid without this restriction.


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

  • [1] R. M. Cohn, Difference algebra, Interscience, New York, 1965. MR 0205987 (34:5812)
  • [2] B. Greenspan, A bound for the orders of the components of a system of algebraic difference equations, Pacific J. Math. 9 (1959), 473-486. MR 0109153 (22:41)
  • [3] E. R. Kolchin, Differential algebra and algebraic groups, Academic Press, New York and London, 1973. MR 0568864 (58:27929)
  • [4] B. A. Lando, Jacobi's bound for the order of systems of first order differential equations, Trans. Amer. Math. Soc. 152 (1970), 119-135. MR 0279079 (43:4805)
  • [5] -, Jacobi's bound for first order difference equations, Proc. Amer. Math. Soc. 32 (1972), 8-12. MR 0289474 (44:6664)
  • [6] J. F. Ritt, Systems of algebraic differential equations, Ann. of Math. (2) 36 (1935), 293-302. MR 1503223
  • [7] -, Jacobi's problem on the order of a system of differential equations, Ann. of Math. (2) 36 (1935), 303-312. MR 1503224
  • [8] J. Tomasovic, A generalized Jacobi conjecture for partial differential equations, Trans. Amor. Math. Soc. (to appear).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 12H05

Retrieve articles in all journals with MSC: 12H05


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0572294-0
Keywords: Order of differential systems, differential kernel, Greenspan bound, Jacobi bound
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society