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

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.

[1]  D. Kapur,et al.  A Completion Procedure for Computing a Canonical Basis for a k-Subalgebra , 1989, Computers and Mathematics.

[2]  David Shannon,et al.  Using Gröbner Bases to Determine Algebra Membership Split Surjective Algebra Homomorphisms Determine Birational Equivalence , 1988, J. Symb. Comput..

[3]  Lorenzo Robbiano,et al.  On the Theory of Graded Structures , 1986, J. Symb. Comput..

[4]  Daniel Lazard,et al.  Resolution des Systemes d'Equations Algebriques , 1981, Theor. Comput. Sci..

[5]  Marc Giusti,et al.  On the castelnuovo regularity for curves , 1989, ISSAC '89.

[6]  N. Bose.  Multidimensional Systems Theory , 1985 .

[7]  Lorenzo Robbiano,et al.  Term Orderings on the Polynominal Ring , 1985, European Conference on Computer Algebra.

[8]  den Essen Arno van.  A criterion to decide if a polynomial map is invertible and to compute the inverse , 1990 .

[9]  Erich Kaltofen,et al.  Computers and Mathematics , 1989, Springer US.

[10]  Ferdinando Mora,et al.  Groebner Bases for Non-Commutative Polynomial Rings , 1985, AAECC.

[11]  Gérard P. Huet,et al.  An Algorithm to Generate the Basis of Solutions to Homogeneous Linear Diophantine Equations , 1978, Inf. Process. Lett..

[12]  F. Ollivier.  Inversibility of rational mappings and structural identifiability in automatics , 1989, ISSAC '89.

[13]  B. Buchberger,et al.  Grobner Bases : An Algorithmic Method in Polynomial Ideal Theory , 1985 .

[14]  H. Michael Möller.  A Reduction Strategy for the Taylor Resolution , 1985, European Conference on Computer Algebra.

[15]  Carlo Traverso,et al.  Computing the conductor of an integral extension , 1991, Discret. Appl. Math..

[16]  Maurice Janet,et al.  Sur les systèmes d'équations aux dérivées partielles , 1920 .

[17]  A. Meyer,et al.  The complexity of the word problems for commutative semigroups and polynomial ideals , 1982 .

[18]  Heinrich W. E. Jung.  Über ganze birationale Transformationen der Ebene. , 1942 .

[19]  H. Bass,et al.  The Jacobian conjecture: Reduction of degree and formal expansion of the inverse , 1982 .