Canonical Bases: Relations with Standard Bases, Finiteness Conditions and Application to Tame Automorphisms

Abstract

Canonical bases for k-subalgeras of k[x 1, …, x n ] 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.