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SubalgebraBases :: SubalgebraBases

SubalgebraBases -- A package for finding canonical subalgebra bases (Sagbi bases)

Description

Let $R=k[f_1,\ldots,f_k]$ denote the subalgebra of the polynomial ring $k[x_1,\ldots,x_n]$ generated by $f_1,\ldots ,f_k.$ We say $f_1,\ldots,f_k$ form a subalgebra basis with respect to a monomial order $<$ if the initial algebra associated to $<$, defined as $in(R) := k[in(f) \mid f \in R],$ is generated by the elements $in(f_1), \ldots , in(f_k).$ The main functions provided by this package are for computing these subalgebra bases: sagbi, subalgebraBasis, and verifySagbi.

Some references for Subalgebra bases (aka canonical subalgebra bases, SAGBI bases)

  • Kapur, D., Madlener, K. (1989). A completion procedure for computing a canonical basis of a $k$-subalgebra. Proceedings of Computers and Mathematics 89 (eds. Kaltofen and Watt), MIT, Cambridge, June 1989
  • Robbiano, L., Sweedler, M. (1990). Subalgebra bases, in W.~Bruns, A.~Simis (eds.): Commutative Algebra, Springer Lecture Notes in Mathematics 1430, pp.~61--87
  • F. Ollivier, Canonical Bases: Relations with Standard bases, finiteness conditions and applications to tame automorphisms, in Effective Methods in Algebraic Geometry, Castiglioncello 1990, pp. 379-400, Progress in Math. 94 Birkhauser, Boston (1991)
  • Stillman, Michael, and Harrison Tsai. Using SAGBI bases to compute invariants. J. Pure and Appl. Alg., 1999, pp.~285--302.
  • B. Sturmfels, Groebner bases and Convex Polytopes, Univ. Lecture Series 8, Amer Math Soc, Providence, 1996

See also

Authors

Version

This documentation describes version 1.1 of SubalgebraBases.

Source code

The source code from which this documentation is derived is in the file SubalgebraBases.m2. The auxiliary files accompanying it are in the directory SubalgebraBases/.

Exports

  • Types
  • Functions and commands
    • groebnerMembershipTest -- Extrinsic method for subring membership
    • isSAGBI -- Check if the generators are a sagbi basis
    • sagbi -- Compute a subalgebra basis (sagbi basis)
    • sagbiBasis -- Constructs a computation object from a subring.
    • subalgebraBasis -- Compute subalgebra basis (sagbi basis) generators
    • subring -- Constructs a subring of a polynomial ring.
    • verifySagbi -- Test if the generators form sagbi basis
  • Methods
    • ambient(SAGBIBasis) -- The ambient ring of a SAGBIBasis computation object
    • ambient(Subring) -- The ambient ring of a subring
    • generators(SAGBIBasis) -- Returns a partial sagbi generating set
    • generators(Subring) -- A generating set of a subring
    • "groebnerMembershipTest(RingElement,Subring)" -- see groebnerMembershipTest -- Extrinsic method for subring membership
    • "isSAGBI(SAGBIBasis)" -- see isSAGBI -- Check if the generators are a sagbi basis
    • "isSAGBI(Subring)" -- see isSAGBI -- Check if the generators are a sagbi basis
    • Matrix % SAGBIBasis -- Remainder modulo a subring
    • Matrix % Subring -- Remainder modulo a subring
    • net(SAGBIBasis) -- Short summary of a sagbi basis computation object
    • net(Subring) -- Short summary of a subring
    • numgens(Subring) -- The number of generators of a subring
    • ring(SAGBIBasis) -- The lifted ring of a SAGBIBasis computation object
    • RingElement % SAGBIBasis -- Remainder modulo a subring
    • RingElement % Subring -- Remainder modulo a subring
    • "sagbi(List)" -- see sagbi -- Compute a subalgebra basis (sagbi basis)
    • "sagbi(Matrix)" -- see sagbi -- Compute a subalgebra basis (sagbi basis)
    • "sagbi(SAGBIBasis)" -- see sagbi -- Compute a subalgebra basis (sagbi basis)
    • "sagbi(Subring)" -- see sagbi -- Compute a subalgebra basis (sagbi basis)
    • "sagbiBasis(HashTable)" -- see sagbiBasis -- Constructs a computation object from a subring.
    • "sagbiBasis(Subring)" -- see sagbiBasis -- Constructs a computation object from a subring.
    • "subalgebraBasis(List)" -- see subalgebraBasis -- Compute subalgebra basis (sagbi basis) generators
    • "subalgebraBasis(Matrix)" -- see subalgebraBasis -- Compute subalgebra basis (sagbi basis) generators
    • "subalgebraBasis(Subring)" -- see subalgebraBasis -- Compute subalgebra basis (sagbi basis) generators
    • "subring(List)" -- see subring -- Constructs a subring of a polynomial ring.
    • "subring(Matrix)" -- see subring -- Constructs a subring of a polynomial ring.
    • "subring(SAGBIBasis)" -- see subring -- Constructs a subring of a polynomial ring.
    • "verifySagbi(List)" -- see verifySagbi -- Test if the generators form sagbi basis
    • "verifySagbi(Matrix)" -- see verifySagbi -- Test if the generators form sagbi basis
    • "verifySagbi(Subring)" -- see verifySagbi -- Test if the generators form sagbi basis
  • Symbols

For the programmer

The object SubalgebraBases is a package.