TY - GEN
T1 - Standard bases of differential ideals
AU - Ollivier, François
N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 1991.
PY - 1991/1/1
Y1 - 1991/1/1
N2 - The aim of this paper is to introduce a new definition of standard bases of differential ideals, allowing more general orderings than the previous one, given by Giuseppa Carrá-Ferro, and following the general definition of standard bases, given in [O3], valid for algebraic ideals, canonical bases of subalgebras, etc. Differential standard bases, as canonical bases, suffer a great limitation: they can be infinite, even for ideals of finite type. Nevertheless, we can sometimes bound the order of intermediate computations, necessary to make some elements of special interest appear in the basis. As an illustration, we consider a differential rational map f: AFn→AFn, and show that if f is birational, then ord f-1 ≤ n ord f. Partial standard bases computations provide then two algorithms to test the existence of f-1. The first one is also able to determine the inverse, if any. The second only determines existence, but we can provide a bound of complexity depending only of n, ord f and the number of derivatives.
AB - The aim of this paper is to introduce a new definition of standard bases of differential ideals, allowing more general orderings than the previous one, given by Giuseppa Carrá-Ferro, and following the general definition of standard bases, given in [O3], valid for algebraic ideals, canonical bases of subalgebras, etc. Differential standard bases, as canonical bases, suffer a great limitation: they can be infinite, even for ideals of finite type. Nevertheless, we can sometimes bound the order of intermediate computations, necessary to make some elements of special interest appear in the basis. As an illustration, we consider a differential rational map f: AFn→AFn, and show that if f is birational, then ord f-1 ≤ n ord f. Partial standard bases computations provide then two algorithms to test the existence of f-1. The first one is also able to determine the inverse, if any. The second only determines existence, but we can provide a bound of complexity depending only of n, ord f and the number of derivatives.
U2 - 10.1007/3-540-54195-0_60
DO - 10.1007/3-540-54195-0_60
M3 - Conference contribution
AN - SCOPUS:0004342708
SN - 9783540541950
T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
SP - 304
EP - 321
BT - Applied Algebra, Algebraic Algorithms and Error-Correcting Codes - 8th International Conference, AAECC-8, Proceedings
A2 - Sakata, Shojiro
PB - Springer Verlag
T2 - 8th International Conference on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC 1990
Y2 - 20 August 1990 through 24 August 1990
ER -