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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

Duval (D.). -- Évaluation dynamique et clôture algébrique en Axiom. Journal of Pure and Applied Algebra, no99, 1995, pp. 267--295.
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.
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.
Haché (G.). -- Construction effective des codes géométriques. -- Thèse de doctorat de l'Université Pierre et Marie Curie (Paris 6), Septembre 1996.
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.
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.
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.
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

(1) -> )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 
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   BlowUpPackage is already explicitly exposed in frame initial 
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      /var/aw/var/LatexWiki/PAFF/spad/BLUPPACK.NRLIB/BLUPPACK
   BlowUpWithQuadTrans is already explicitly exposed in frame initial 
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      /var/aw/var/LatexWiki/PAFF/spad/BLQT.NRLIB/BLQT
   BlowUpMethodCategory is already explicitly exposed in frame initial 
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      initial 
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      from /var/aw/var/LatexWiki/PAFF/spad/BLHN.NRLIB/BLHN
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   AffineSpace is already explicitly exposed in frame initial 
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fricas

-- First we define the field GF(2).

K:=PF 2

$\label{eq1}\hbox{\axiomType{PrimeField}\ } \left({2}\right)$

(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)

$\label{eq2}\hbox{\axiomType{DistributedMultivariatePolynomial}\ } \left({{\left[ X , \: Y , \: Z \right]}, \:{\hbox{\axiomType{PrimeField}\ } \left({2}\right)}}\right)$

(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)

$\label{eq3}\hbox{\axiomType{PackageForAlgebraicFunctionField}\ } \left({{\hbox{\axiomType{PrimeField}\ } \left({2}\right)}, \:{\left[ X , \: Y , \: Z \right]}, \: \hbox{\axiomType{BlowUpWithQuadTrans}\ }}\right)$

(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 !!

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