login  home  contents  what's new  discussion  bug reports     help  links  subscribe  changes  refresh  edit

PAFF : Package for Algebraic Function Fields in one variable

by Gaétan Haché

PAFF is a package written in Axiom and one of its many purpose is to construct geometric Goppa codes (also called algebraic geometric codes or AG-codes). I wrote this package in the frame work of my doctorate thesis on "Effective construction of geometric codes": this thesis was done at Inria in Rocquencourt at project CODES and under the direction of Dominique LeBrigand? at Université Pierre et Marie Curie (Paris 6). Here is a résumé of my thesis.

It is well known that the most difficult part in constructing AG-code is the computation of a basis of the vector space "L(D)" where D is a divisor of the function field of an irreducible curve. To compute such a basis, PAFF used the Brill-Noether algorithm which was generalized to any plane curve by D. LeBrigand and J.J. Risler (see [7] ). In [4] you will find more details about the algorithmic aspect of the Brill-Noether algorithm. Also, if you prefer, as I do, a strictly algebraic approach, see [3]. This is the approach I used in my thesis ([4]) and of course this is where you will find complete details about the implementation of the algorithm. The algebraic approach use the theory of algebraic function field in one variable : you will find in [8] a very good introduction to this theory and AG-codes.

It is important to notice that PAFF can be used for most computation related to the function field of an irreducible plane curve. For example, you can compute the genus, find all places above all the singular points, compute the adjunction divisor and of course compute a basis of the vector space L(D) for any divisor D of the function field of the curve.

There is also the package PAFFFF which is especially designed to be used over finite fields. This package is essentially the same as PAFF, except that the computation are done over "dynamic extensions" of the ground field. For this, I used a simplify version of the notion of dynamic algebraic closure as proposed by D. Duval (see [1]).

References

  1. Duval (D.). -- Évaluation dynamique et clôture algébrique en Axiom. Journal of Pure and Applied Algebra, no99, 1995, pp. 267--295.
  2. Garcia (A.) et Stichtenoth (H.). -- A tower of Artin-Schreier extensions of function fields attaining the Drinfeld-Vladut bound. Invent. Math., vol. 121, 1995, pp. 211--222.
  3. Haché (G.). -- Computation in algebraic function fields for effective construction of algebraic-geometric codes. Lecture Notes in Computer Science, vol. 948, 1995, pp. 262--278.
  4. Haché (G.). -- Construction effective des codes géométriques. -- Thèse de doctorat de l'Université Pierre et Marie Curie (Paris 6), Septembre 1996.
  5. Haché (G.) et Le Brigand (D.). -- Effective construction of algebraic geometry codes. IEEE Transaction on Information Theory, vol. 41, n'27 6, November 1995, pp. 1615--1628.
  6. Huang (M.D.) et Ierardi (D.). -- Efficient algorithms for Riemann-Roch problem and for addition in the jacobian of a curve. In: Proceedings 32nd Annual Symposium on Foundations of Computer Sciences. IEEE Comput. Soc. Press, pp. 678--687.
  7. Le Brigand (D.) et Risler (J.J.). -- Algorithme de Brill-Noether et codes de Goppa. Bull. Soc. Math. France, vol. 116, 1988, pp. 231--253.
  8. Stichtenoth (H.). -- Algebraic function fields and codes. -- Springer-Verlag, 1993, University Text.

Example 1

This example compute the genus of the projective plane curve defined by:

       5    2 3      4
      X  + Y Z  + Y Z  = 0

over the field GF(2).

First load the PAFF library (must be done twice).

fricas
)lib )dir PAFF/spad
UnivariateTaylorSeriesCZero is now explicitly exposed in frame initial UnivariateTaylorSeriesCZero will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/UTSZ.NRLIB/UTSZ SetCategoryWithDegree is now explicitly exposed in frame initial SetCategoryWithDegree will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/SETCATD.NRLIB/SETCATD RootsFindingPackage is now explicitly exposed in frame initial RootsFindingPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/RFP.NRLIB/RFP ProjectiveSpaceCategory is now explicitly exposed in frame initial ProjectiveSpaceCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PRSPCAT.NRLIB/PRSPCAT ProjectiveSpace is now explicitly exposed in frame initial ProjectiveSpace will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PROJSP.NRLIB/PROJSP ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField is now explicitly exposed in frame initial ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PROJPLPS.NRLIB/PROJPLPS ProjectivePlane is now explicitly exposed in frame initial ProjectivePlane will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PROJPL.NRLIB/PROJPL ProjectiveAlgebraicSetPackage is now explicitly exposed in frame initial ProjectiveAlgebraicSetPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PRJALGPK.NRLIB/PRJALGPK PolynomialPackageForCurve is now explicitly exposed in frame initial
PolynomialPackageForCurve will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLPKCRV.NRLIB/PLPKCRV Plcs is now explicitly exposed in frame initial Plcs will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLCS.NRLIB/PLCS PlacesOverPseudoAlgebraicClosureOfFiniteField is now explicitly exposed in frame initial PlacesOverPseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLACESPS.NRLIB/PLACESPS PlacesCategory is now explicitly exposed in frame initial PlacesCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLACESC.NRLIB/PLACESC Places is now explicitly exposed in frame initial Places will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLACES.NRLIB/PLACES PackageForPoly is now explicitly exposed in frame initial PackageForPoly will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PFORP.NRLIB/PFORP ParametrizationPackage is now explicitly exposed in frame initial ParametrizationPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PARAMP.NRLIB/PARAMP PackageForAlgebraicFunctionFieldOverFiniteField is now explicitly exposed in frame initial PackageForAlgebraicFunctionFieldOverFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PAFFFF.NRLIB/PAFFFF PackageForAlgebraicFunctionField is now explicitly exposed in frame initial PackageForAlgebraicFunctionField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PAFF.NRLIB/PAFF PseudoAlgebraicClosureOfRationalNumberCategory is now explicitly exposed in frame initial PseudoAlgebraicClosureOfRationalNumberCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACRATC.NRLIB/PACRATC PseudoAlgebraicClosureOfRationalNumber is now explicitly exposed in frame initial PseudoAlgebraicClosureOfRationalNumber will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACRAT.NRLIB/PACRAT PseudoAlgebraicClosureOfPerfectFieldCategory is now explicitly exposed in frame initial PseudoAlgebraicClosureOfPerfectFieldCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACPERC.NRLIB/PACPERC PseudoAlgebraicClosureOfFiniteField is now explicitly exposed in frame initial PseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACOFF.NRLIB/PACOFF PseudoAlgebraicClosureOfFiniteFieldCategory is now explicitly exposed in frame initial PseudoAlgebraicClosureOfFiniteFieldCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACFFC.NRLIB/PACFFC PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory is now explicitly exposed in frame initial PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACEXTC.NRLIB/PACEXTC PseudoAlgebraicClosureOfAlgExtOfRationalNumber is now explicitly exposed in frame initial PseudoAlgebraicClosureOfAlgExtOfRationalNumber will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACEXT.NRLIB/PACEXT NeitherSparseOrDensePowerSeries is now explicitly exposed in frame initial NeitherSparseOrDensePowerSeries will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/NSDPS.NRLIB/NSDPS NewtonPolygon is now explicitly exposed in frame initial NewtonPolygon will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/NPOLYGON.NRLIB/NPOLYGON LocalParametrizationOfSimplePointPackage is now explicitly exposed in frame initial LocalParametrizationOfSimplePointPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/LPARSPT.NRLIB/LPARSPT LinesOpPack is now explicitly exposed in frame initial LinesOpPack will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/LOP.NRLIB/LOP LocalPowerSeriesCategory is now explicitly exposed in frame initial LocalPowerSeriesCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/LOCPOWC.NRLIB/LOCPOWC LinearSystemFromPowerSeriesPackage is now explicitly exposed in frame initial LinearSystemFromPowerSeriesPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/LISYSER.NRLIB/LISYSER InterpolateFormsPackage is now explicitly exposed in frame initial InterpolateFormsPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INTFRSP.NRLIB/INTFRSP InterfaceGroebnerPackage is now explicitly exposed in frame initial InterfaceGroebnerPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INTERGB.NRLIB/INTERGB IntersectionDivisorPackage is now explicitly exposed in frame initial IntersectionDivisorPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INTDIVP.NRLIB/INTDIVP InfinitlyClosePoint is now explicitly exposed in frame initial InfinitlyClosePoint will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INFCLSPT.NRLIB/INFCLSPT InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField is now explicitly exposed in frame initial InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INFCLSPS.NRLIB/INFCLSPS InfinitlyClosePointCategory is now explicitly exposed in frame initial InfinitlyClosePointCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INFCLCT.NRLIB/INFCLCT InfClsPt is now explicitly exposed in frame initial InfClsPt will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/ICP.NRLIB/ICP GeneralPackageForAlgebraicFunctionField is now explicitly exposed in frame initial GeneralPackageForAlgebraicFunctionField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/GPAFF.NRLIB/GPAFF FiniteFieldSquareFreeDecomposition is now explicitly exposed in frame initial FiniteFieldSquareFreeDecomposition will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/FFSQFR.NRLIB/FFSQFR FiniteFieldFactorizationWithSizeParseBySideEffect is now explicitly exposed in frame initial FiniteFieldFactorizationWithSizeParseBySideEffect will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/FFFACTSE.NRLIB/FFFACTSE FactorisationOverPseudoAlgebraicClosureOfRationalNumber is now explicitly exposed in frame initial FactorisationOverPseudoAlgebraicClosureOfRationalNumber will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/FACTRN.NRLIB/FACTRN FactorisationOverPseudoAlgebraicClosureOfAlgExtOfRationalNumber is now explicitly exposed in frame initial FactorisationOverPseudoAlgebraicClosureOfAlgExtOfRationalNumber will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/FACTEXT.NRLIB/FACTEXT DesingTreePackage is now explicitly exposed in frame initial DesingTreePackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DTP.NRLIB/DTP DesingTree is now explicitly exposed in frame initial DesingTree will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DSTREE.NRLIB/DSTREE DesingTreeCategory is now explicitly exposed in frame initial DesingTreeCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DSTRCAT.NRLIB/DSTRCAT DivisorCategory is now explicitly exposed in frame initial DivisorCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DIVCAT.NRLIB/DIVCAT Divisor is now explicitly exposed in frame initial Divisor will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DIV.NRLIB/DIV BlowUpPackage is now explicitly exposed in frame initial BlowUpPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/BLUPPACK.NRLIB/BLUPPACK BlowUpWithQuadTrans is now explicitly exposed in frame initial BlowUpWithQuadTrans will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/BLQT.NRLIB/BLQT BlowUpMethodCategory is now explicitly exposed in frame initial BlowUpMethodCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/BLMETCT.NRLIB/BLMETCT BlowUpWithHamburgerNoether is now explicitly exposed in frame initial BlowUpWithHamburgerNoether will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/BLHN.NRLIB/BLHN AffineSpaceCategory is now explicitly exposed in frame initial AffineSpaceCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFSPCAT.NRLIB/AFSPCAT AffineSpace is now explicitly exposed in frame initial AffineSpace will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFFSP.NRLIB/AFFSP AffinePlaneOverPseudoAlgebraicClosureOfFiniteField is now explicitly exposed in frame initial AffinePlaneOverPseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFFPLPS.NRLIB/AFFPLPS AffinePlane is now explicitly exposed in frame initial AffinePlane will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFFPL.NRLIB/AFFPL AffineAlgebraicSetComputeWithResultant is now explicitly exposed in frame initial AffineAlgebraicSetComputeWithResultant will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFALGRES.NRLIB/AFALGRES AffineAlgebraicSetComputeWithGroebnerBasis is now explicitly exposed in frame initial AffineAlgebraicSetComputeWithGroebnerBasis will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFALGGRO.NRLIB/AFALGGRO
fricas
)lib )dir PAFF/spad
UnivariateTaylorSeriesCZero is already explicitly exposed in frame initial UnivariateTaylorSeriesCZero will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/UTSZ.NRLIB/UTSZ SetCategoryWithDegree is already explicitly exposed in frame initial
SetCategoryWithDegree will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/SETCATD.NRLIB/SETCATD RootsFindingPackage is already explicitly exposed in frame initial RootsFindingPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/RFP.NRLIB/RFP ProjectiveSpaceCategory is already explicitly exposed in frame initial ProjectiveSpaceCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PRSPCAT.NRLIB/PRSPCAT ProjectiveSpace is already explicitly exposed in frame initial ProjectiveSpace will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PROJSP.NRLIB/PROJSP ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField is already explicitly exposed in frame initial ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PROJPLPS.NRLIB/PROJPLPS ProjectivePlane is already explicitly exposed in frame initial ProjectivePlane will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PROJPL.NRLIB/PROJPL ProjectiveAlgebraicSetPackage is already explicitly exposed in frame initial ProjectiveAlgebraicSetPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PRJALGPK.NRLIB/PRJALGPK PolynomialPackageForCurve is already explicitly exposed in frame initial PolynomialPackageForCurve will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLPKCRV.NRLIB/PLPKCRV Plcs is already explicitly exposed in frame initial Plcs will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLCS.NRLIB/PLCS PlacesOverPseudoAlgebraicClosureOfFiniteField is already explicitly exposed in frame initial PlacesOverPseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLACESPS.NRLIB/PLACESPS PlacesCategory is already explicitly exposed in frame initial PlacesCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLACESC.NRLIB/PLACESC Places is already explicitly exposed in frame initial Places will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PLACES.NRLIB/PLACES PackageForPoly is already explicitly exposed in frame initial PackageForPoly will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PFORP.NRLIB/PFORP ParametrizationPackage is already explicitly exposed in frame initial ParametrizationPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PARAMP.NRLIB/PARAMP PackageForAlgebraicFunctionFieldOverFiniteField is already explicitly exposed in frame initial PackageForAlgebraicFunctionFieldOverFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PAFFFF.NRLIB/PAFFFF PackageForAlgebraicFunctionField is already explicitly exposed in frame initial PackageForAlgebraicFunctionField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PAFF.NRLIB/PAFF PseudoAlgebraicClosureOfRationalNumberCategory is already explicitly exposed in frame initial PseudoAlgebraicClosureOfRationalNumberCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACRATC.NRLIB/PACRATC PseudoAlgebraicClosureOfRationalNumber is already explicitly exposed in frame initial PseudoAlgebraicClosureOfRationalNumber will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACRAT.NRLIB/PACRAT PseudoAlgebraicClosureOfPerfectFieldCategory is already explicitly exposed in frame initial PseudoAlgebraicClosureOfPerfectFieldCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACPERC.NRLIB/PACPERC PseudoAlgebraicClosureOfFiniteField is already explicitly exposed in frame initial PseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACOFF.NRLIB/PACOFF PseudoAlgebraicClosureOfFiniteFieldCategory is already explicitly exposed in frame initial PseudoAlgebraicClosureOfFiniteFieldCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACFFC.NRLIB/PACFFC PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory is already explicitly exposed in frame initial PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACEXTC.NRLIB/PACEXTC PseudoAlgebraicClosureOfAlgExtOfRationalNumber is already explicitly exposed in frame initial PseudoAlgebraicClosureOfAlgExtOfRationalNumber will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/PACEXT.NRLIB/PACEXT NeitherSparseOrDensePowerSeries is already explicitly exposed in frame initial NeitherSparseOrDensePowerSeries will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/NSDPS.NRLIB/NSDPS NewtonPolygon is already explicitly exposed in frame initial NewtonPolygon will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/NPOLYGON.NRLIB/NPOLYGON LocalParametrizationOfSimplePointPackage is already explicitly exposed in frame initial LocalParametrizationOfSimplePointPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/LPARSPT.NRLIB/LPARSPT LinesOpPack is already explicitly exposed in frame initial LinesOpPack will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/LOP.NRLIB/LOP LocalPowerSeriesCategory is already explicitly exposed in frame initial LocalPowerSeriesCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/LOCPOWC.NRLIB/LOCPOWC LinearSystemFromPowerSeriesPackage is already explicitly exposed in frame initial LinearSystemFromPowerSeriesPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/LISYSER.NRLIB/LISYSER InterpolateFormsPackage is already explicitly exposed in frame initial InterpolateFormsPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INTFRSP.NRLIB/INTFRSP InterfaceGroebnerPackage is already explicitly exposed in frame initial InterfaceGroebnerPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INTERGB.NRLIB/INTERGB IntersectionDivisorPackage is already explicitly exposed in frame initial IntersectionDivisorPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INTDIVP.NRLIB/INTDIVP InfinitlyClosePoint is already explicitly exposed in frame initial InfinitlyClosePoint will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INFCLSPT.NRLIB/INFCLSPT InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField is already explicitly exposed in frame initial InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INFCLSPS.NRLIB/INFCLSPS InfinitlyClosePointCategory is already explicitly exposed in frame initial InfinitlyClosePointCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/INFCLCT.NRLIB/INFCLCT InfClsPt is already explicitly exposed in frame initial InfClsPt will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/ICP.NRLIB/ICP GeneralPackageForAlgebraicFunctionField is already explicitly exposed in frame initial GeneralPackageForAlgebraicFunctionField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/GPAFF.NRLIB/GPAFF FiniteFieldSquareFreeDecomposition is already explicitly exposed in frame initial FiniteFieldSquareFreeDecomposition will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/FFSQFR.NRLIB/FFSQFR FiniteFieldFactorizationWithSizeParseBySideEffect is already explicitly exposed in frame initial FiniteFieldFactorizationWithSizeParseBySideEffect will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/FFFACTSE.NRLIB/FFFACTSE FactorisationOverPseudoAlgebraicClosureOfRationalNumber is already explicitly exposed in frame initial FactorisationOverPseudoAlgebraicClosureOfRationalNumber will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/FACTRN.NRLIB/FACTRN FactorisationOverPseudoAlgebraicClosureOfAlgExtOfRationalNumber is already explicitly exposed in frame initial FactorisationOverPseudoAlgebraicClosureOfAlgExtOfRationalNumber will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/FACTEXT.NRLIB/FACTEXT DesingTreePackage is already explicitly exposed in frame initial DesingTreePackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DTP.NRLIB/DTP DesingTree is already explicitly exposed in frame initial DesingTree will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DSTREE.NRLIB/DSTREE DesingTreeCategory is already explicitly exposed in frame initial DesingTreeCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DSTRCAT.NRLIB/DSTRCAT DivisorCategory is already explicitly exposed in frame initial DivisorCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DIVCAT.NRLIB/DIVCAT Divisor is already explicitly exposed in frame initial Divisor will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/DIV.NRLIB/DIV BlowUpPackage is already explicitly exposed in frame initial BlowUpPackage will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/BLUPPACK.NRLIB/BLUPPACK BlowUpWithQuadTrans is already explicitly exposed in frame initial BlowUpWithQuadTrans will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/BLQT.NRLIB/BLQT BlowUpMethodCategory is already explicitly exposed in frame initial BlowUpMethodCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/BLMETCT.NRLIB/BLMETCT BlowUpWithHamburgerNoether is already explicitly exposed in frame initial BlowUpWithHamburgerNoether will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/BLHN.NRLIB/BLHN AffineSpaceCategory is already explicitly exposed in frame initial AffineSpaceCategory will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFSPCAT.NRLIB/AFSPCAT AffineSpace is already explicitly exposed in frame initial AffineSpace will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFFSP.NRLIB/AFFSP AffinePlaneOverPseudoAlgebraicClosureOfFiniteField is already explicitly exposed in frame initial AffinePlaneOverPseudoAlgebraicClosureOfFiniteField will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFFPLPS.NRLIB/AFFPLPS AffinePlane is already explicitly exposed in frame initial AffinePlane will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFFPL.NRLIB/AFFPL AffineAlgebraicSetComputeWithResultant is already explicitly exposed in frame initial AffineAlgebraicSetComputeWithResultant will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFALGRES.NRLIB/AFALGRES AffineAlgebraicSetComputeWithGroebnerBasis is already explicitly exposed in frame initial AffineAlgebraicSetComputeWithGroebnerBasis will be automatically loaded when needed from /var/aw/var/LatexWiki/PAFF/spad/AFALGGRO.NRLIB/AFALGGRO

fricas
-- First we define the field GF(2).
K:=PF 2

\label{eq1}\hbox{\axiomType{PrimeField}\ } (2)(1)
Type: Type
fricas
-- Next, we define the polynomial ring over which
-- the polynomial is defined. 
-- You have the choice for the name of 
-- the three variables (always three !!) but  
-- the domain  DMP must be used. 
-- DMP  is an AXIOM domain and stands for DistributedMultivariatePolymnomial.
R:=DMP([X,Y,Z],K)

\label{eq2}\hbox{\axiomType{DistributedMultivariatePolynomial}\ } ([ X , Y , Z ] , \hbox{\axiomType{PrimeField}\ } (2))(2)
Type: Type
fricas
-- Then we tell to the package PAFF over which field the computation must be done.
-- Also, you must give the same list of variables which is used to defined the polynomial.
-- BLQT Stand for BlowUpWithQuadTrans which specified the method
-- used for blowing-up (there will be another one (when ?) 
-- using similar thechnics to Hamburger-Nother expansions).
P:=PAFF(K,[X,Y,Z],BLQT)

\label{eq3}\hbox{\axiomType{PackageForAlgebraicFunctionField}\ } (\hbox{\axiomType{PrimeField}\ } (2) , [ X , Y , Z ] , \hbox{\axiomType{BlowUpWithQuadTrans}\ })(3)
Type: Type
fricas
-- We defined now the polynomial of the curve.
C:R:=X**5 + Y**2*Z**3+Y*Z**4
There are no library operations named ** Use HyperDoc Browse or issue )what op ** to learn if there is any operation containing " ** " in its name.
Cannot find a definition or applicable library operation named ** with argument type(s) DistributedMultivariatePolynomial([X,Y,Z],PrimeField(2)) PositiveInteger
Perhaps you should use "@" to indicate the required return type, or "$" to specify which version of the function you need.




  Subject:   Be Bold !!
  ( 13 subscribers )  
Please rate this page: