JAS - Gröbner bases book

# Gröbner bases book algorithms and JAS methods

Summary of algorithms from the Gröbner bases book and corresponding JAS classes and methods.

## Gröbner bases book algorithms

The JAS base package `edu.jas` name is omitted in the following table. JAS also contains improved versions of the algorithms which may be located through the links. A short explanation of code organization with interfaces and several implementing classes can be found in the API guide.

GB book algorithm | JAS interfaces, classes and methods | remarks
0.1 `DIVINT` | `arith.BigInteger.divideAndRemainder` | adapter for `java.math.BigInteger.divideAndRemainder`
2.1 `DIV` | `structure.MonoidElem.divide` and `structure.MonoidElem.remainder` | all classes which implement this interface
2.2 `DIVPOL` | `poly.GenPolynomial.` `divideAndRemainder` | for univariate polynomials over fields
2.3 `EXTEUC` | `poly.GenPolynomial.egcd` | for univariate polynomials over fields
3.1 `LINDEP` | `` not implemented | But see `vector.LinAlg.` `rowEchelonForm` and other classes in the `vector` package.
3.2 `EXCHANGE` | `` not implemented |
4.1 `EQUIV` | `structure.Residue.equals` or `application.Residue.equals` | arbitrary residue class rings and residue class rings modulo polynomial ideals
5.1 `REDPOL` | `gb.Reduction.normalform`, `gb.ReductionAbstract.normalform`, `gb.ReductionSeq.normalform` | interface and sequential computation class, the method exists also for polynomial lists and with a reduction recording matrix (this is exactly `REDPOL`)
5.2 `REDUCTION` | `gb.Reduction.irreducibleSet` |
5.3 `GRÖBNERTEST` | `gb.GroebnerBase.isGB` | provided by all classes which implement this interface
5.4 `GRÖBNER` | not implemented |
5.5 `REDGRÖBNER` | `gb.GroebnerBase.GB` with `gb.OrderedMinPairlist` | provided by all classes which implement the interface and allow the selection of the pair-list in a constructor
5.6 `GRÖBNERNEW1` | `gb.GroebnerBase.GB` with `gb.OrderedPairlist` | provided by all classes which implement the interface and allow the selection of the pair-list in a constructor
5.7 `UPDATE`, 5.8 `GRÖBNERNEW2` | `gb.GroebnerBase.GB` with `gb.OrderedSyzPairlist` | provided by all classes which implement the interface and allow the selection of the pair-list in a constructor
5.9 `EXTGRÖBNER` | `gb.GroebnerBase.extGB` | provided by some classes which implement the interface
6.1 `ELIMINATION` | `application.Ideal.eliminate` | the version with the `String[]` parameter computes a Gröbner base wrt. the corresponding elimination order
6.2 `PROPER` | `application.Ideal.isONE` | proper ideal test is `! id.isONE()`
6.3 `INTERSECTION` | `application.Ideal.intersect` | for lists of ideals a simple iterative algorithm is used and for a pair of ideals it is the same algorithm
6.4 `CRT` | `gbufd.PolyGBUtil.chineseRemainderTheorem` |
6.5 `IDEALDIV1` | `application.Ideal.quotient` | for an ideal a simple iterative algorithm is used and for one polynomial it is an algorithm without computing syzygies
6.6 `IDEALDIV2` | `application.Ideal.` `infiniteQuotientRab` and `application.Ideal.` `infiniteQuotientExponent` | the exponent is computed in a separate step at the moment
6.7 `RADICALMEMTEST` | `application.Ideal.isRadicalMember` | the exponent is not computed
6.8 `SUBRINGMEMTEST` | `gbufd.PolyGBUtil.subRingAndMember` |
8.1 `PREDEC` | `application.Ideal.` `zeroDimDecomposition` | univariate polynomials of minimal degree in the ideal are irreducible and not a power of an irreducible polynomial as specified in `PREDEC`
8.2 `ZRADICALTEST` | `application.Ideal.isZeroDimRadical` | will also work in characteritsic p > 0
8.3 `ZRADICAL` | `application.Ideal.radical` | `ZRADICAL` is containted as special case, see also `application.Ideal.` `zeroDimRadicalDecomposition`
8.4 `NORMPRIMDEC` | `application.Ideal.` `zeroDimPrimaryDecomposition` | contains all preprocessing steps, see `ZPRIMDEC`
8.5 `NORMPOS` | `application.Ideal.normalPositionFor` | one step of `NORMPOS` as explained for the modified algorithm on page 383ff
8.6 `ZPRIMDEC` | `application.Ideal.` `zeroDimPrimaryDecomposition` | returns a list of `PrimaryComponent` containers
8.7 `CONT` | `application.Ideal.contraction` | more complicated since the permutation of variables must be considered also, see `application.Ideal.permContraction`, the polynomial `f` is returned in the `IdealWithUniv` container
8.8 `EXTCONT` | `application.Ideal.extension` | only `EXT`, the combination `EXTCONT` is not implemented explicitly, the polynomial `f` is returned in the `IdealWithUniv` container
8.9 `RADICAL` | `application.Ideal.radical` | for ideals with arbitrary dimension
8.10 `PRIMDEC` | `application.Ideal.` `primaryDecomposition` | for ideals with arbitrary dimension
8.11 `VARSIGN` | `root.RootUtil.signVar` |
8.12 `STURMSEQ` | `root.RealRootsSturm.sturmSequence` |
8.13 `ISOLATE` | `root.RealRoots.realRoots` | could also be used for real root isolations not using Sturm sequences
8.14 `ISOREC` | `root.RealRootsSturm.realRoots` | interval bi-section until a root is isolated
8.15 `ISOREFINE` | `root.RealRoots.refineInterval` | also for real root isolations not using Sturm sequences
8.16 `SQUEEZE` | `root.RealRoots.` `invariantMagnitudeInterval` | see also `root.RealRoots.` `realMagnitude`
8.17 `REALZEROS` | `application.PolyUtilApp.` `realAlgebraicRoots` | different algorithm and different result data structure with `RealAlgebraicNumbers`
9.1 `REDTERMS` | `gbufd.GroebnerBaseFGLM.redTerms` |
9.2 `UNIVPOL` | `application.Ideal.` `constructUnivariate` |
9.3 `CONVGRÖBNER` | `gbufd.GroebnerBaseFGLM.` `convGroebnerToLex` | See also `gbufd.GroebnerBaseFGLM.GB`. It computes a Gröbner base with respect to a graded term order and then uses FGLM to convert to a lexicographic term order. See also `gbufd.GroebnerBaseWalk.GB`. It computes a Gröbner base with respect to a term order `t1` and then uses Gröbner Walk to convert to a GB with term order `t2`.
9.4 `LMINTERM` | `gbufd.GroebnerBaseFGLM.lMinterm` |
9.5 `STRCONST` | `` not implemented |
9.6 `DIMENSION` | `application.Ideal.dimension` |
10.1 `D-GRÖBNER` | `gb.DGroebnerBaseSeq.GB` and `gb.EGroebnerBaseSeq.GB` |

Last modified: Thu Jul 8 17:47:59 CEST 2021