|Title||Canonical Bases: Relation with Standard Bases, Finiteness Conditions and
Application to Tame Automorphisms|
|Author(s)|| Francois Ollivier|
|Text||F. Ollivier. Canonical Bases: Relation with Standard Bases, Finiteness
Conditions and Application to Tame Automorphisms, In Proceedings of MEGA '90 , Castinglioncello, Italy, Birkhauser, 1990.|
|Type||Technical Report, Misc|
|Abstract||Canonical bases for k-subalgeras of k[x 1 ; : : : ; xn ] are analogs of standard bases for ideals. They form a set of generators, which allows to answer the membership problem by a reduction process. Unfortunately, they may be infinite even for finitely generated subalgeras. We redefine|
canonical bases, and for that we recall some properties of monoids, k-algebras of monoids and "binomial" ideals, which play an essential role in our presentation and the implementation we made in the IBM computer algebra system Scratchpad II. We complete the already known relations between standard bases and canonical bases by generalizing the notion of standard bases for ideals of any k-subalgebra admitting a finite canonical basis. We also have a way of finding a set of generators of the ideal of relations between elements of a canonical basis, which is a standard basis for some ordering.
We then turn to finiteness conditions, and investigate the case of integrally closed subalgebras. We show that if some integral extension B of a subalgebra A admits a finite canonical basis, we have an algorithm to solve the membership problem for A, by computing the generalized standard basis of a B-ideal. We conjecture that any integrally closed subalgebra admits a finite canonical basis, and provide partial results.