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PAFF

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"PAFF":http://www-rocq.inria.fr/codes/Gaetan.Hache/PAFF.html
: 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":http://www-rocq.inria.fr/codes/Gaetan.Hache/these/resumeThese.html
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).

\begin{axiom}
)lib )dir PAFF/spad
)lib )dir PAFF/spad
\end{axiom}

\begin{axiom}

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

K:=PF 2  

-- 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)

-- 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)

-- We defined now the polynomial of the curve.

C:R:=X**5 + Y**2*Z**3+Y*Z**4

-- We give it to the package PAFF(K,[X,Y,Z]) which was assigned to the
-- variable "P"


setCurve(C)$P

-- To compute the genus of the curve, simply do 


genus()$P


-- To compute the genus, the package use the genus formula 
-- given by the blowin-up theory. That means that the singular points
-- has been computed. 

singularPoints()$P

-- The results of singularPoints()$P is the list of all
-- the singular points of the curve in the projective plane.
--
--
-- The Brill-Noether algorithm use the notion of "adjunction divisor".
-- To compute it simply do "adjunctionDivisor()$P"
-- You should obtained something like
--
--            1
--       8 %I1
--
-- This is a divisor of the function field of the curve, consisting
-- of 8 times the place %I1 which is of degree 1 (the exponant).
-- The place %I1 is a place above a singular point (the unique one for this example).
-- This mean that the "desingularization tree" has been computed. See next step.

adjunctionDivisor()$P


-- To compute the "desingularization tree" simply do:
-- desingTree()$P
--
-- For this example, you should obtained from desingTree()$P
--
--        ["UU.."]
--
-- This a list of desingularization tree for each singular point.
-- Here there is only one, which is "UU..".  
-- To interpret the result, you have to do some manual drawing.
-- The letter U means "Up", and a . (dot) means "down".
-- 
--
--
--              o
--              |  
--           ^  |  |
--           |  |  | .
--         U |  |  | 
--           |  |  v
--              |
--              |
--              o
--              |  
--           ^  |  |
--           |  |  | .
--         U |  |  |
--           |  |  v
--              |
--              |
--              o
--         

desingTree()$P

-- To see more information about the desingularization trees,
-- issue the command, fullDesTree()$P, and recall the command 
-- desingTree()$P. Here you have a bit more information about
-- the infinitly near points in the desingularization trees.
-- For this example, the result corresponds to the following
--
--         %I1  o  multiplicity = 1
--              |  
--              |
--              |
--              |
--              |
--              |
--              |
--         %I0  o  multiplicity = 2
--              |  
--              |
--              |
--              |
--              |
--              |
--              |
--         %P0  o  multiplicity = 3
--         


fullDesTree()$P
desingTree()$P


-- To see everything about desingularization trees, issue
-- the following

fullInfClsPt()$P
--desingTree()$P


-- You can ask for all the place of degree 1
--

placesOfDegree(1)$P


-- With those places, you can create divisors 


listOfDiv:=placesOfDegree(1)$P :: List DIV PLACES PF 2

-- You can add the divisors.

D:=reduce(+, listOfDiv)

-- You can multiply the divisor by an integer

D10 := 10 * D


-- You can ask for the degree of the divisor

degree D10


-- You can compute the basis of the vector space L(D10).
-- The results is an Axiom Record. The first selector "num"
-- corresponds to the numerators of the elements of the basis,
-- and the second selector "den" is the common denominator.


baseOfLofD:= lBasis(D10)$P

-- Of course, the number of element in the list of numerator
-- is the dimension of the vector space L(D10). According to the 
-- Riemann-Roch Theorem, since 
-- 
--  deg D10 >= 2 g - 1
-- 
-- we should have 
--
-- dim L(D10) = deg D10 - g + 1
--
--

((# baseOfLofD.num) = degree D10 - genus()$P + 1 ) :: Boolean

\end{axiom}

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).

axiom

)lib )dir PAFF/spad

   UnivariateTaylorSeriesCZero is now explicitly exposed in frame
      initial
   UnivariateTaylorSeriesCZero will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/UTSZ.NRLIB/code
   SetCategoryWithDegree is now explicitly exposed in frame initial
   SetCategoryWithDegree will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/SETCATD.NRLIB/code
   RootsFindingPackage is now explicitly exposed in frame initial
   RootsFindingPackage will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/RFP.NRLIB/code
   ProjectiveSpaceCategory is now explicitly exposed in frame initial
   ProjectiveSpaceCategory will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/PRSPCAT.NRLIB/code
   ProjectiveSpace is now explicitly exposed in frame initial
   ProjectiveSpace will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PROJSP.NRLIB/code
   ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField is now
      explicitly exposed in frame initial
   ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField will be
      automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PROJPLPS.NRLIB/code
   ProjectivePlane is now explicitly exposed in frame initial
   ProjectivePlane will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PROJPL.NRLIB/code
   ProjectiveAlgebraicSetPackage is now explicitly exposed in frame
      initial
   ProjectiveAlgebraicSetPackage will be automatically loaded when
      needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PRJALGPK.NRLIB/code
   PolynomialPackageForCurve is now explicitly exposed in frame initial

   PolynomialPackageForCurve will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/PLPKCRV.NRLIB/code
   Plcs is now explicitly exposed in frame initial
   Plcs will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PLCS.NRLIB/code
   PlacesOverPseudoAlgebraicClosureOfFiniteField is now explicitly
      exposed in frame initial
   PlacesOverPseudoAlgebraicClosureOfFiniteField will be automatically
      loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PLACESPS.NRLIB/code
   PlacesCategory is now explicitly exposed in frame initial
   PlacesCategory will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PLACESC.NRLIB/code
   Places is now explicitly exposed in frame initial
   Places will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PLACES.NRLIB/code
   PackageForPoly is now explicitly exposed in frame initial
   PackageForPoly will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PFORP.NRLIB/code
   ParametrizationPackage is now explicitly exposed in frame initial
   ParametrizationPackage will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PARAMP.NRLIB/code
   PackageForAlgebraicFunctionFieldOverFiniteField is now explicitly
      exposed in frame initial
   PackageForAlgebraicFunctionFieldOverFiniteField will be
      automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PAFFFF.NRLIB/code
   PackageForAlgebraicFunctionField is now explicitly exposed in frame
      initial
   PackageForAlgebraicFunctionField will be automatically loaded when
      needed from /var/zope2/var/LatexWiki/PAFF/spad/PAFF.NRLIB/code
   PseudoAlgebraicClosureOfRationalNumberCategory is now explicitly
      exposed in frame initial
   PseudoAlgebraicClosureOfRationalNumberCategory will be automatically
      loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PACRATC.NRLIB/code
   PseudoAlgebraicClosureOfRationalNumber is now explicitly exposed in
      frame initial
   PseudoAlgebraicClosureOfRationalNumber will be automatically loaded
      when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PACRAT.NRLIB/code
   PseudoAlgebraicClosureOfPerfectFieldCategory is now explicitly
      exposed in frame initial
   PseudoAlgebraicClosureOfPerfectFieldCategory will be automatically
      loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PACPERC.NRLIB/code
   PseudoAlgebraicClosureOfFiniteField is now explicitly exposed in
      frame initial
   PseudoAlgebraicClosureOfFiniteField will be automatically loaded
      when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PACOFF.NRLIB/code
   PseudoAlgebraicClosureOfFiniteFieldCategory is now explicitly
      exposed in frame initial
   PseudoAlgebraicClosureOfFiniteFieldCategory will be automatically
      loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PACFFC.NRLIB/code
   PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory is now
      explicitly exposed in frame initial
   PseudoAlgebraicClosureOfAlgExtOfRationalNumberCategory will be
      automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PACEXTC.NRLIB/code
   PseudoAlgebraicClosureOfAlgExtOfRationalNumber is now explicitly
      exposed in frame initial
   PseudoAlgebraicClosureOfAlgExtOfRationalNumber will be automatically
      loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PACEXT.NRLIB/code
   NeitherSparseOrDensePowerSeries is now explicitly exposed in frame
      initial
   NeitherSparseOrDensePowerSeries will be automatically loaded when
      needed from /var/zope2/var/LatexWiki/PAFF/spad/NSDPS.NRLIB/code
   NewtonPolygon is now explicitly exposed in frame initial
   NewtonPolygon will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/NPOLYGON.NRLIB/code
   LocalParametrizationOfSimplePointPackage is now explicitly exposed
      in frame initial
   LocalParametrizationOfSimplePointPackage will be automatically
      loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/LPARSPT.NRLIB/code
   LinesOpPack is now explicitly exposed in frame initial
   LinesOpPack will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/LOP.NRLIB/code
   LocalPowerSeriesCategory is now explicitly exposed in frame initial
   LocalPowerSeriesCategory will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/LOCPOWC.NRLIB/code
   LinearSystemFromPowerSeriesPackage is now explicitly exposed in
      frame initial
   LinearSystemFromPowerSeriesPackage will be automatically loaded when
      needed from /var/zope2/var/LatexWiki/PAFF/spad/LISYSER.NRLIB/code
   InterpolateFormsPackage is now explicitly exposed in frame initial
   InterpolateFormsPackage will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/INTFRSP.NRLIB/code
   InterfaceGroebnerPackage is now explicitly exposed in frame initial
   InterfaceGroebnerPackage will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/INTERGB.NRLIB/code
   IntersectionDivisorPackage is now explicitly exposed in frame
      initial
   IntersectionDivisorPackage will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/INTDIVP.NRLIB/code
   InfinitlyClosePoint is now explicitly exposed in frame initial
   InfinitlyClosePoint will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/INFCLSPT.NRLIB/code
   InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField is now
      explicitly exposed in frame initial
   InfinitlyClosePointOverPseudoAlgebraicClosureOfFiniteField will be
      automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/INFCLSPS.NRLIB/code
   InfinitlyClosePointCategory is now explicitly exposed in frame
      initial
   InfinitlyClosePointCategory will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/INFCLCT.NRLIB/code
   InfClsPt is now explicitly exposed in frame initial
   InfClsPt will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/ICP.NRLIB/code
   GeneralPackageForAlgebraicFunctionField is now explicitly exposed in
      frame initial
   GeneralPackageForAlgebraicFunctionField will be automatically loaded
      when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/GPAFF.NRLIB/code
   FiniteFieldSquareFreeDecomposition is now explicitly exposed in
      frame initial
   FiniteFieldSquareFreeDecomposition will be automatically loaded when
      needed from /var/zope2/var/LatexWiki/PAFF/spad/FFSQFR.NRLIB/code
   FiniteFieldFactorizationWithSizeParseBySideEffect is now explicitly
      exposed in frame initial
   FiniteFieldFactorizationWithSizeParseBySideEffect will be
      automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/FFFACTSE.NRLIB/code
   FactorisationOverPseudoAlgebraicClosureOfRationalNumber is now
      explicitly exposed in frame initial
   FactorisationOverPseudoAlgebraicClosureOfRationalNumber will be
      automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/FACTRN.NRLIB/code
   FactorisationOverPseudoAlgebraicClosureOfAlgExtOfRationalNumber is
      now explicitly exposed in frame initial
   FactorisationOverPseudoAlgebraicClosureOfAlgExtOfRationalNumber will
      be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/FACTEXT.NRLIB/code
   DesingTreePackage is now explicitly exposed in frame initial
   DesingTreePackage will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/DTP.NRLIB/code
   DesingTree is now explicitly exposed in frame initial
   DesingTree will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/DSTREE.NRLIB/code
   DesingTreeCategory is now explicitly exposed in frame initial
   DesingTreeCategory will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/DSTRCAT.NRLIB/code
   DivisorCategory is now explicitly exposed in frame initial
   DivisorCategory will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/DIVCAT.NRLIB/code
   Divisor is now explicitly exposed in frame initial
   Divisor will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/DIV.NRLIB/code
   BlowUpPackage is now explicitly exposed in frame initial
   BlowUpPackage will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/BLUPPACK.NRLIB/code
   BlowUpWithQuadTrans is now explicitly exposed in frame initial
   BlowUpWithQuadTrans will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/BLQT.NRLIB/code
   BlowUpMethodCategory is now explicitly exposed in frame initial
   BlowUpMethodCategory will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/BLMETCT.NRLIB/code
   BlowUpWithHamburgerNoether is now explicitly exposed in frame
      initial
   BlowUpWithHamburgerNoether will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/BLHN.NRLIB/code
   AffineSpaceCategory is now explicitly exposed in frame initial
   AffineSpaceCategory will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/AFSPCAT.NRLIB/code
   AffineSpace is now explicitly exposed in frame initial
   AffineSpace will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/AFFSP.NRLIB/code
   AffinePlaneOverPseudoAlgebraicClosureOfFiniteField is now explicitly
      exposed in frame initial
   AffinePlaneOverPseudoAlgebraicClosureOfFiniteField will be
      automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/AFFPLPS.NRLIB/code
   AffinePlane is now explicitly exposed in frame initial
   AffinePlane will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/AFFPL.NRLIB/code
   AffineAlgebraicSetComputeWithResultant is now explicitly exposed in
      frame initial
   AffineAlgebraicSetComputeWithResultant will be automatically loaded
      when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/AFALGRES.NRLIB/code
   AffineAlgebraicSetComputeWithGroebnerBasis is now explicitly exposed
      in frame initial
   AffineAlgebraicSetComputeWithGroebnerBasis will be automatically
      loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/AFALGGRO.NRLIB/code

axiom

)lib )dir PAFF/spad

   UnivariateTaylorSeriesCZero is already explicitly exposed in frame
      initial
   UnivariateTaylorSeriesCZero will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/UTSZ.NRLIB/code
   SetCategoryWithDegree is already explicitly exposed in frame initial

   SetCategoryWithDegree will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/SETCATD.NRLIB/code
   RootsFindingPackage is already explicitly exposed in frame initial
   RootsFindingPackage will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/RFP.NRLIB/code
   ProjectiveSpaceCategory is already explicitly exposed in frame
      initial
   ProjectiveSpaceCategory will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/PRSPCAT.NRLIB/code
   ProjectiveSpace is already explicitly exposed in frame initial
   ProjectiveSpace will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PROJSP.NRLIB/code
   ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField is already
      explicitly exposed in frame initial
   ProjectivePlaneOverPseudoAlgebraicClosureOfFiniteField will be
      automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PROJPLPS.NRLIB/code
   ProjectivePlane is already explicitly exposed in frame initial
   ProjectivePlane will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PROJPL.NRLIB/code
   ProjectiveAlgebraicSetPackage is already explicitly exposed in frame
      initial
   ProjectiveAlgebraicSetPackage will be automatically loaded when
      needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PRJALGPK.NRLIB/code
   PolynomialPackageForCurve is already explicitly exposed in frame
      initial
   PolynomialPackageForCurve will be automatically loaded when needed
      from /var/zope2/var/LatexWiki/PAFF/spad/PLPKCRV.NRLIB/code
   Plcs is already explicitly exposed in frame initial
   Plcs will be automatically loaded when needed from
      /var/zope2/var/LatexWiki/PAFF/spad/PLCS.NRLIB/code
   PlacesOverPseudoAlgebraicClosureOfFiniteField is already explicitly
      exposed in frame initial
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axiom

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

K:=PF 2

(1)

Type: Domain

axiom

-- 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)

(2)

Type: Domain

axiom

-- 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)

(3)

Type: Domain

axiom

-- We defined now the polynomial of the curve.

C:R:=X**5 + Y**2*Z**3+Y*Z**4

(4)

Type: DistributedMultivariatePolynomial?([X,Y,Z]?,PrimeField? 2)

axiom

-- We give it to the package PAFF(K,[X,Y,Z]) which was assigned to the
-- variable "P"

setCurve(C)$P

(5)

Type: DistributedMultivariatePolynomial?([X,Y,Z]?,PrimeField? 2)

axiom

-- To compute the genus of the curve, simply do 

genus()$P

(6)

Type: NonNegativeInteger

axiom

-- To compute the genus, the package use the genus formula
-- given by the blowin-up theory. That means that the singular points
-- has been computed. 

singularPoints()$P

(7)

Type: List ProjectivePlane? PrimeField? 2

axiom

-- The results of singularPoints()$P is the list of all
-- the singular points of the curve in the projective plane.
--
--
-- The Brill-Noether algorithm use the notion of "adjunction divisor".
-- To compute it simply do "adjunctionDivisor()$P"
-- You should obtained something like
--
--            1
--       8 %I1
--
-- This is a divisor of the function field of the curve, consisting
-- of 8 times the place %I1 which is of degree 1 (the exponant).
-- The place %I1 is a place above a singular point (the unique one for this example).
-- This mean that the "desingularization tree" has been computed. See next step.

adjunctionDivisor()$P

(8)

Type: Divisor Places PrimeField? 2

axiom

-- To compute the "desingularization tree" simply do:
-- desingTree()$P
--
-- For this example, you should obtained from desingTree()$P
--
--        ["UU.."]
--
-- This a list of desingularization tree for each singular point.
-- Here there is only one, which is "UU..".
-- To interpret the result, you have to do some manual drawing.
-- The letter U means "Up", and a . (dot) means "down".
--
--
--
--              o
--              |
--           ^  |  |
--           |  |  | .
--         U |  |  |
--           |  |  v
--              |
--              |
--              o
--              |
--           ^  |  |
--           |  |  | .
--         U |  |  |
--           |  |  v
--              |
--              |
--              o
--         

desingTree()$P

(9)

Type: List DesingTree? InfClsPt?(PrimeField? 2,[X,Y,Z]?,BlowUpWithQuadTrans?)

axiom

-- To see more information about the desingularization trees,
-- issue the command, fullDesTree()$P, and recall the command
-- desingTree()$P. Here you have a bit more information about
-- the infinitly near points in the desingularization trees.
-- For this example, the result corresponds to the following
--
--         %I1  o  multiplicity = 1
--              |
--              |
--              |
--              |
--              |
--              |
--              |
--         %I0  o  multiplicity = 2
--              |
--              |
--              |
--              |
--              |
--              |
--              |
--         %P0  o  multiplicity = 3
--         

fullDesTree()$P

Type: Void

axiom

desingTree()$P

(10)

Type: List DesingTree? InfClsPt?(PrimeField? 2,[X,Y,Z]?,BlowUpWithQuadTrans?)

axiom

-- To see everything about desingularization trees, issue
-- the following

fullInfClsPt()$P

Type: Void

axiom

--desingTree()$P

-- You can ask for all the place of degree 1
--

placesOfDegree(1)$P

(11)

Type: List Places PrimeField? 2

axiom

-- With those places, you can create divisors 

listOfDiv:=placesOfDegree(1)$P :: List DIV PLACES PF 2

(12)

Type: List Divisor Places PrimeField? 2

axiom

-- You can add the divisors.

D:=reduce(+, listOfDiv)

(13)

Type: Divisor Places PrimeField? 2

axiom

-- You can multiply the divisor by an integer

D10 := 10 * D

(14)

Type: Divisor Places PrimeField? 2

axiom

-- You can ask for the degree of the divisor

degree D10

(15)

Type: PositiveInteger

axiom

-- You can compute the basis of the vector space L(D10).
-- The results is an Axiom Record. The first selector "num"
-- corresponds to the numerators of the elements of the basis,
-- and the second selector "den" is the common denominator.

baseOfLofD:= lBasis(D10)$P

   Trying to interpolate with forms of degree:
   8
   Denominator found
   Intersection Divisor of Denominator found

(16)

Type: Record(num: List DistributedMultivariatePolynomial?([X,Y,Z]?,PrimeField? 2),den: DistributedMultivariatePolynomial?([X,Y,Z]?,PrimeField? 2))

axiom

-- Of course, the number of element in the list of numerator
-- is the dimension of the vector space L(D10). According to the
-- Riemann-Roch Theorem, since
--
--  deg D10 >= 2 g - 1
--
-- we should have
--
-- dim L(D10) = deg D10 - g + 1
--
--

((# baseOfLofD.num) = degree D10 - genus()$P + 1 ) :: Boolean

(17)

Type: Boolean