Geodesic
On standard bases in rings of differential polynomials
Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 3, pp. 89-102

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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 $\beta$-orderings) that contains only one element. Various properties of admissible orderings on differential monomials are studied. We bring up the following problem: whether there is a finitely generated differential ideal that does not admit a finite standard basis w.r.t. any ordering. 
@article{FPM_2003_9_3_a5,
     author = {A. I. Zobnin},
     title = {On standard bases in rings of differential polynomials},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
     pages = {89--102},
     publisher = {mathdoc},
     volume = {9},
     number = {3},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a5/}
}
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A. I. Zobnin. On standard bases in rings of differential polynomials. Fundamentalʹnaâ i prikladnaâ matematika, Tome 9 (2003) no. 3, pp. 89-102. http://geodesic.mathdoc.fr/item/FPM_2003_9_3_a5/