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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 isSAGBI.

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 1.3 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
  • Methods
    • ambient(SAGBIBasis) -- The ambient ring of a SAGBIBasis computation object
    • ambient(Subring) -- The ambient ring of a subring
    • "forceSB(SAGBIBasis)" -- see forceSB -- declare the generators of a subring or SAGBIBasis to be a complete sagbi basis
    • "forceSB(Subring)" -- see forceSB -- declare the generators of a subring or SAGBIBasis to be a complete sagbi basis
    • 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
    • "groebnerSubductionQuotient(RingElement,Subring)" -- see groebnerSubductionQuotient -- Extrinsic method for subduction quotients
    • "isSAGBI(List)" -- see isSAGBI -- Check if the generators are a sagbi basis
    • "isSAGBI(Matrix)" -- see isSAGBI -- Check if the generators are a sagbi basis
    • "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(SAGBIBasis) -- The number of generators of a SAGBIBasis
    • 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
    • RingElement // Subring -- subductionQuotient with respect to 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.
    • "sagbiDegree(SAGBIBasis)" -- see sagbiDegree -- The current degree of the sagbi computation
    • "sagbiLimit(SAGBIBasis)" -- see sagbiLimit -- The current limit of the sagbi computation
    • "sagbiStatus(SAGBIBasis)" -- see sagbiStatus -- returns if the sagbi computation is done
    • status(SAGBIBasis) -- status of the sagbi computation
    • "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
    • "subduction(List,List)" -- see subduction -- Subduction of polynomials
    • "subduction(List,RingElement)" -- see subduction -- Subduction of polynomials
    • "subduction(Matrix,Matrix)" -- see subduction -- Subduction of polynomials
    • "subduction(Matrix,RingElement)" -- see subduction -- Subduction of polynomials
    • "subduction(SAGBIBasis,Matrix)" -- see subduction -- Subduction of polynomials
    • "subduction(SAGBIBasis,RingElement)" -- see subduction -- Subduction of polynomials
    • "subduction(Subring,Matrix)" -- see subduction -- Subduction of polynomials
    • "subduction(Subring,RingElement)" -- see subduction -- Subduction of polynomials
    • "subductionQuotientRing(Subring)" -- see subductionQuotientRing -- returns the subduction quotient ring of a subring
    • "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.
    • "subringIntersection(Subring,Subring)" -- see subringIntersection -- Intersection of subrings
  • Symbols
    • AutoSubduce -- Flag for autosubduction before the Sagbi algorithm
    • AutoSubduceOnPartialCompletion -- Subduct sagbi generators at the end of the Sagbi algorithm
    • Compute -- Flag for performing computations while checking a sagbi basis
    • GeneratorSymbol -- variables for the subductionQuotientRing
    • PrintLevel -- Levels of information displayed during Sagbi algorithm
    • Recompute -- Flag for restarting a sagbi computation
    • ReduceNewGenerators -- Flag for reducing new generators in Sagbi algorithm
    • RenewOptions -- Flag for reselecting the options for a sagbi computation
    • StorePending -- Flag for storing the pending list to the result of the Sagbi algorithm
    • SubductionMethod -- Subduction method for the Sagbi algorithm
    • UseSubringGens -- Flag for using the subring generators when checking a sagbi basis

For the programmer

The object SubalgebraBases is a package.