The Greenspan bound for the order of differential systems
Author:
Richard M. Cohn
Journal:
Proc. Amer. Math. Soc. 79 (1980), 523526
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 YorkLondonSydeny, 1965. MR 0205987
(34 #5812)
 [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
(22 #41)
 [3]
E.
R. Kolchin, Differential algebra and algebraic groups,
Academic Press, New YorkLondon, 1973. Pure and Applied Mathematics, Vol.
54. MR
0568864 (58 #27929)
 [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
(43 #4805), http://dx.doi.org/10.1090/S00029947197002790791
 [5]
Barbara
A. Lando, Jacobi’s bound for first order
difference equations, Proc. Amer. Math.
Soc. 32 (1972),
8–12. MR
0289474 (44 #6664), http://dx.doi.org/10.1090/S0002993919720289474X
 [6]
J.
F. Ritt, Systems of algebraic differential equations, Ann. of
Math. (2) 36 (1935), no. 2, 293–302. MR
1503223, http://dx.doi.org/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, http://dx.doi.org/10.2307/1968572
 [8]
J. Tomasovic, A generalized Jacobi conjecture for partial differential equations, Trans. Amor. Math. Soc. (to appear).
 [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), 473486. 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), 119135. MR 0279079 (43:4805)
 [5]
 , Jacobi's bound for first order difference equations, Proc. Amer. Math. Soc. 32 (1972), 812. MR 0289474 (44:6664)
 [6]
 J. F. Ritt, Systems of algebraic differential equations, Ann. of Math. (2) 36 (1935), 293302. MR 1503223
 [7]
 , Jacobi's problem on the order of a system of differential equations, Ann. of Math. (2) 36 (1935), 303312. MR 1503224
 [8]
 J. Tomasovic, A generalized Jacobi conjecture for partial differential equations, Trans. Amor. Math. Soc. (to appear).
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198005722940
PII:
S 00029939(1980)05722940
Keywords:
Order of differential systems,
differential kernel,
Greenspan bound,
Jacobi bound
Article copyright:
© Copyright 1980
American Mathematical Society
