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.
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.
There is a simple case, but of special interest, where the complexity of computing a canonical basis is known: the case where k[f 1, …, f n ] = k[x 1, …, x n ]. We show that the canonical bases procedure give more information than previously known methods and may provide a tool for the tame generators conjecture.
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Ollivier, F. (1991). Canonical Bases: Relations with Standard Bases, Finiteness Conditions and Application to Tame Automorphisms. In: Mora, T., Traverso, C. (eds) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol 94. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0441-1_25
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DOI: https://doi.org/10.1007/978-1-4612-0441-1_25
Publisher Name: Birkhäuser, Boston, MA
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