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Canonical bases: Relations with standard bases, finiteness conditions and application to tame automorphisms. (English) Zbl 0734.13015
Effective methods in algebraic geometry, Proc. Symp., Castiglioncello/Italy 1990, Prog. Math. 94, 379-400 (1991).
[For the entire collection see Zbl 0721.00009.]
Let k be a field, P the algebra of polynomials in n variables over k and choose a term-ordering \(<\) on the set M of monomials in P. For a subset S of P define 1t(S) to be the set of leading-terms (with respect to \(<)\) of all nonzero elements in S. If A is a subalgebra of P then 1t(A) is a submonoid of M. - A subset E of A is a “canonical basis” of A if 1t(E) generates the monoid 1t(A). So canonical bases for subalgebras of P are the analoga of Gröbner bases for ideals in P.
The author describes an algorithm (implemented in Scratchpad II) for the computation of a canonical basis of a subalgebra of P. He presents the necessary background material, discusses several related questions and applies the theory of canonical bases to study automorphisms of P.

13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
14H37 Automorphisms of curves