## Abstract

We consider Ollivier’s standard bases (also known as differential Gröbner bases) in an ordinary ring of differential polynomials in one indeterminate. We establish a link between these bases and Levi’s reduction process. We prove that the ideal [*x*
^{p}] has a finite standard basis (w.r.t. the so-called *β*-orderings) that contains only one element. Various properties of admissible orderings on differential monomials are studied. We bring up the question of whether there is a finitely generated differential ideal that does not admit a finite standard basis w.r.t. any ordering.

This is a preview of subscription content, access via your institution.

## References

- 1.
F. Boulier, D. Lazard, F. Ollivier, and M. Petitot, “Representation for the radical of a finitely generated differential ideal,” in:

*Proceedings of 1995 International Symposium on Symbolic and Algebraic Computation*, ACM Press (1995), pp. 158–166. - 2.
G. Carrà Ferro, “Gröbner bases and differential algebra,” in:

*Lecture Notes in Computer Science*, Vol. 356, 1989, pp. 141–150. - 3.
G. Carrà Ferro, “Differential Gröbner bases in one variable and in the partial case,” in:

*Math. Comput. Modelling*, Vol. 25, Pergamon Press (1997), pp. 1–10. - 4.
G. Gallo, B. Mishra, and F. Ollivier, “Some constructions in rings of differential polynomials,” in:

*Lecture Notes in Computer Science*, Vol. 539, 1991, pp. 171–182. - 5.
E. Hubert, “Factorization-free decomposition algorithms in differential algebra,”

*J. Symb. Comp.*,**29**, 641–662 (2000). - 6.
E. R. Kolchin,

*Differential Algebra and Algebraic Groups*, Academic Press (1973). - 7.
H. Levi, “On the structure of differential polynomials and on their theory of ideals,”

*Trans. Amer. Math. Soc.*,**51**, 532–568 (1942). - 8.
D. G. Mead, “A necessary and sufficient condition for membership in [

*uv*],”*Proc. Amer. Math. Soc.*,**17**, 470–473 (1966). - 9.
D. G. Mead and M. E. Newton, “Syzygies in [

*ypz*],”*Proc. Amer. Math. Soc.*,**43**, No. 2, 301–305 (1974). - 10.
K. B. O’Keefe, “A property of the differential ideal [

*yp*],”*Trans. Amer. Math. Soc.*,**94**, 483–497 (1960). - 11.
F. Ollivier,

*Le problème de l’identifiabilité structurelle globale*, Doctoral Dissertation, Paris (1990). - 12.
F. Ollivier, “Standard bases of differential ideals,” in:

*Lecture Notes in Computer Science*, Vol. 508, 304–321 (1990). - 13.
A. Ovchinnikov and A. Zobnin, “Classification and applications of monomial orderings and the properties of differential orderings,” in: V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, eds.,

*Proceedings of the Fifth International Workshop on Computer Algebra in Scientific Computing (CASC-2002)*, Technische Universität München, Garching, Germany (2002), pp. 237–252. - 14.
E. V. Pankratiev, “Some approaches to construction of standard bases in commutative and differential algebra,” in: V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, eds.,

*Proceedings of the Fifth International Workshop on Computer Algebra in Scientific Computing (CASC-2002)*, Technische Universität München, Garching, Germany (2002), pp. 265–268. - 15.
E. V. Pankratiev, “Some approaches to construction of the differential Gröbner bases,” in:

*Calculemus 2002. 10th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning. Marseille, France, July 3–5, 2002. Work in Progress Papers*, Univ. Saarlandes (2002), pp. 50–55. - 16.
J. F. Ritt,

*Differential Algebra*, Volume XXXIII of Colloquium Publications, American Mathematical Society, New York (1950). - 17.
C. Rust and G. J. Reid, “Rankings of partial derivatives,” in:

*Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation*, ACM Press, New York (1997), pp. 9–16. - 18.
V. Weispfenning, “Differential term-orders,” in:

*Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation*, ACM Press, Kiev (1993), pp. 245–253. - 19.
A. Zobnin, “Essential properties of admissible orderings and rankings,” to appear in

*Contributions to General Algebra*,**14**(2004). Available at http://shade.msu.ru/~difalg/Articles/Our/Zobnin/ess-properties.ps.

## Author information

### Affiliations

## Additional information

__________

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 9, No. 3, pp. 89–102, 2003.

## Rights and permissions

## About this article

### Cite this article

Zobnin, A.I. On standard bases in rings of differential polynomials.
*J Math Sci* **135, **3327–3335 (2006). https://doi.org/10.1007/s10958-006-0161-3

Issue Date:

### Keywords

- Reduction Process
- Standard Basis
- Differential Polynomial
- Differential Ideal
- Admissible Ordering