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

MathSciNet review:
572294

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Abstract: Let *S* be a system of ordinary differential polynomials in indeterminates and of order at most in . It was shown by J. F. Ritt that if is a component of *S* of differential dimension 0, then the order of is at most . 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.

**[1]**Richard M. Cohn,*Difference algebra*, Interscience Publishers John Wiley & Sons, New York-London-Sydeny, 1965. MR**0205987****[2]**Bernard 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****[3]**E. R. Kolchin,*Differential algebra and algebraic groups*, Academic Press, New York-London, 1973. Pure and Applied Mathematics, Vol. 54. MR**0568864****[4]**Barbara 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**, 10.1090/S0002-9947-1970-0279079-1**[5]**Barbara A. Lando,*Jacobi’s bound for first order difference equations*, Proc. Amer. Math. Soc.**32**(1972), 8–12. MR**0289474**, 10.1090/S0002-9939-1972-0289474-X**[6]**J. F. Ritt,*Systems of algebraic differential equations*, Ann. of Math. (2)**36**(1935), no. 2, 293–302. MR**1503223**, 10.2307/1968571**[7]**J. F. Ritt,*Jacobi’s problem on the order of a system of differential equations*, Ann. of Math. (2)**36**(1935), no. 2, 303–312. MR**1503224**, 10.2307/1968572**[8]**J. Tomasovic,*A generalized Jacobi conjecture for partial differential equations*, Trans. Amor. Math. Soc. (to appear).

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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