path: root/src/algebra/divisor.spad.pamphlet

\documentclass{article}
\usepackage{open-axiom}
\begin{document}
\title{\$SPAD/src/algebra divisor.spad}
\author{Manuel Bronstein}
\maketitle
\begin{abstract}
\end{abstract}
\eject
\tableofcontents
\eject
\section{domain FRIDEAL FractionalIdeal}
<<domain FRIDEAL FractionalIdeal>>=
)abbrev domain FRIDEAL FractionalIdeal
++ Author: Manuel Bronstein
++ Date Created: 27 Jan 1989
++ Date Last Updated: 30 July 1993
++ Keywords: ideal, algebra, module.
++ Examples: )r FRIDEAL INPUT
++ Description: Fractional ideals in a framed algebra.
FractionalIdeal(R, F, UP, A): Exports == Implementation where
  R : EuclideanDomain
  F : QuotientFieldCategory R
  UP: UnivariatePolynomialCategory F
  A : Join(FramedAlgebra(F, UP), RetractableTo F)

  VF  ==> Vector F
  VA  ==> Vector A
  UPA ==> SparseUnivariatePolynomial A
  QF  ==> Fraction UP

  Exports ==> Group with
    ideal   : VA -> %
      ++ ideal([f1,...,fn]) returns the ideal \spad{(f1,...,fn)}.
    basis   : %  -> VA
      ++ basis((f1,...,fn)) returns the vector \spad{[f1,...,fn]}.
    norm    : %  -> F
      ++ norm(I) returns the norm of the ideal I.
    numer   : %  -> VA
      ++ numer(1/d * (f1,...,fn)) = the vector \spad{[f1,...,fn]}.
    denom   : %  -> R
      ++ denom(1/d * (f1,...,fn)) returns d.
    minimize: %  -> %
      ++ minimize(I) returns a reduced set of generators for \spad{I}.
    randomLC: (NonNegativeInteger, VA) -> A
      ++ randomLC(n,x) should be local but conditional.

  Implementation ==> add
    import CommonDenominator(R, F, VF)
    import MatrixCommonDenominator(UP, QF)
    import InnerCommonDenominator(R, F, List R, List F)
    import MatrixCategoryFunctions2(F, Vector F, Vector F, Matrix F,
                        UP, Vector UP, Vector UP, Matrix UP)
    import MatrixCategoryFunctions2(UP, Vector UP, Vector UP,
                        Matrix UP, F, Vector F, Vector F, Matrix F)
    import MatrixCategoryFunctions2(UP, Vector UP, Vector UP,
                        Matrix UP, QF, Vector QF, Vector QF, Matrix QF)

    Rep := Record(num:VA, den:R)

    poly    : % -> UPA
    invrep  : Matrix F -> A
    upmat   : (A, NonNegativeInteger) -> Matrix UP
    summat  : % -> Matrix UP
    num2O   : VA -> OutputForm
    agcd    : List A -> R
    vgcd    : VF -> R
    mkIdeal : (VA, R) -> %
    intIdeal: (List A, R) -> %
    ret?    : VA -> Boolean
    tryRange: (NonNegativeInteger, VA, R, %) -> Union(%, "failed")

    1               == [[1]$VA, 1]
    numer i         == i.num
    denom i         == i.den
    mkIdeal(v, d)   == [v, d]
    invrep m        == represents(transpose(m) * coordinates(1$A))
    upmat(x, i)     == map(monomial(#1, i)$UP, regularRepresentation x)
    ret? v          == any?(retractIfCan(#1)@Union(F,"failed") case F, v)
    x = y           == denom(x) = denom(y) and numer(x) = numer(y)
    agcd l  == reduce("gcd", [vgcd coordinates a for a in l]$List(R), 0)

    norm i ==
      ("gcd"/[retract(u)@R for u in coefficients determinant summat i])
              / denom(i) ** rank()$A

    tryRange(range, nm, nrm, i) ==
      for j in 0..10 repeat
        a := randomLC(10 * range, nm)
        unit? gcd((retract(norm a)@R exquo nrm)::R, nrm) =>
                                return intIdeal([nrm::F::A, a], denom i)
      "failed"

    summat i ==
      m := minIndex(v := numer i)
      reduce("+",
            [upmat(qelt(v, j + m), j) for j in 0..#v-1]$List(Matrix UP))

    inv i ==
      m  := inverse(map(#1::QF, summat i))::Matrix(QF)
      cd  := splitDenominator(denom(i)::F::UP::QF * m)
      cd2 := splitDenominator coefficients(cd.den)
      invd:= cd2.den / reduce("gcd", cd2.num)
      d   := reduce("max", [degree p for p in parts(cd.num)])
      ideal
        [invd * invrep map(coefficient(#1, j), cd.num) for j in 0..d]$VA

    ideal v ==
      d := reduce("lcm", [commonDenominator coordinates qelt(v, i)
                          for i in minIndex v .. maxIndex v]$List(R))
      intIdeal([d::F * qelt(v, i) for i in minIndex v .. maxIndex v], d)

    intIdeal(l, d) ==
      lr := empty()$List(R)
      nr := empty()$List(A)
      for x in removeDuplicates l repeat
        if (u := retractIfCan(x)@Union(F, "failed")) case F
          then lr := concat(retract(u::F)@R, lr)
          else nr := concat(x, nr)
      r    := reduce("gcd", lr, 0)
      g    := agcd nr
      a    := (r quo (b := gcd(gcd(d, r), g)))::F::A
      d    := d quo b
      r ~= 0 and ((g exquo r) case R) => mkIdeal([a], d)
      invb := inv(b::F)
      va:VA := [invb * m for m in nr]
      zero? a => mkIdeal(va, d)
      mkIdeal(concat(a, va), d)

    vgcd v ==
      reduce("gcd",
             [retract(v.i)@R for i in minIndex v .. maxIndex v]$List(R))

    poly i ==
      m := minIndex(v := numer i)
      +/[monomial(qelt(v, i + m), i) for i in 0..#v-1]

    i1 * i2 ==
      intIdeal(coefficients(poly i1 * poly i2), denom i1 * denom i2)

    i:$ ** m:Integer ==
      negative? m => inv(i) ** (-m)
      n := m::NonNegativeInteger
      v := numer i
      intIdeal([qelt(v, j) ** n for j in minIndex v .. maxIndex v],
               denom(i) ** n)

    num2O v ==
      paren [qelt(v, i)::OutputForm
             for i in minIndex v .. maxIndex v]$List(OutputForm)

    basis i ==
      v := numer i
      d := inv(denom(i)::F)
      [d * qelt(v, j) for j in minIndex v .. maxIndex v]

    coerce(i:$):OutputForm ==
      nm := num2O numer i
      one? denom i => nm
      (1::Integer::OutputForm) / (denom(i)::OutputForm) * nm

    if F has Finite then
      randomLC(m, v) ==
        +/[random()$F * qelt(v, j) for j in minIndex v .. maxIndex v]
    else
      randomLC(m, v) ==
        +/[(random()$Integer rem m::Integer) * qelt(v, j)
            for j in minIndex v .. maxIndex v]

    minimize i ==
      n := (#(nm := numer i))
      one?(n) or (n < 3 and ret? nm) => i
      nrm    := retract(norm mkIdeal(nm, 1))@R
      for range in 1..5 repeat
        (u := tryRange(range, nm, nrm, i)) case $ => return(u::$)
      i

@
\section{package FRIDEAL2 FractionalIdealFunctions2}
<<package FRIDEAL2 FractionalIdealFunctions2>>=
)abbrev package FRIDEAL2 FractionalIdealFunctions2
++ Lifting of morphisms to fractional ideals.
++ Author: Manuel Bronstein
++ Date Created: 1 Feb 1989
++ Date Last Updated: 27 Feb 1990
++ Keywords: ideal, algebra, module.
FractionalIdealFunctions2(R1, F1, U1, A1, R2, F2, U2, A2):
 Exports == Implementation where
  R1, R2: EuclideanDomain
  F1: QuotientFieldCategory R1
  U1: UnivariatePolynomialCategory F1
  A1: Join(FramedAlgebra(F1, U1), RetractableTo F1)
  F2: QuotientFieldCategory R2
  U2: UnivariatePolynomialCategory F2
  A2: Join(FramedAlgebra(F2, U2), RetractableTo F2)

  Exports ==> with
    map: (R1 -> R2, FractionalIdeal(R1, F1, U1, A1)) ->
                                         FractionalIdeal(R2, F2, U2, A2)
	++ map(f,i) \undocumented{}

  Implementation ==> add
    fmap: (F1 -> F2, A1) -> A2

    fmap(f, a) ==
      v := coordinates a
      represents
        [f qelt(v, i) for i in minIndex v .. maxIndex v]$Vector(F2)

    map(f, i) ==
      b := basis i
      ideal [fmap(f(numer #1) / f(denom #1), qelt(b, j))
             for j in minIndex b .. maxIndex b]$Vector(A2)

@
\section{package MHROWRED ModularHermitianRowReduction}
<<package MHROWRED ModularHermitianRowReduction>>=
)abbrev package MHROWRED ModularHermitianRowReduction
++ Modular hermitian row reduction.
++ Author: Manuel Bronstein
++ Date Created: 22 February 1989
++ Date Last Updated: 24 November 1993
++ Keywords: matrix, reduction.
-- should be moved into matrix whenever possible
ModularHermitianRowReduction(R): Exports == Implementation where
  R: EuclideanDomain

  Z   ==> Integer
  V   ==> Vector R
  M   ==> Matrix R
  REC ==> Record(val:R, cl:Z, rw:Z)

  Exports ==> with
    rowEch       : M -> M
      ++ rowEch(m) computes a modular row-echelon form of m, finding
      ++ an appropriate modulus.
    rowEchelon   : (M, R) -> M
      ++ rowEchelon(m, d) computes a modular row-echelon form mod d of
      ++    [d     ]
      ++    [  d   ]
      ++    [    . ]
      ++    [     d]
      ++    [   M  ]
      ++ where \spad{M = m mod d}.
    rowEchLocal    : (M, R) -> M
      ++ rowEchLocal(m,p) computes a modular row-echelon form of m, finding
      ++ an appropriate modulus over a local ring where p is the only prime.
    rowEchelonLocal: (M, R, R) -> M
      ++ rowEchelonLocal(m, d, p) computes the row-echelon form of m
      ++ concatenated with d times the identity matrix
      ++ over a local ring where p is the only prime.
    normalizedDivide: (R, R) -> Record(quotient:R, remainder:R)
      ++ normalizedDivide(n,d) returns a normalized quotient and
      ++ remainder such that consistently unique representatives
      ++ for the residue class are chosen, e.g. positive remainders



  Implementation ==> add
    order   : (R, R) -> Z
    vconc   : (M, R) -> M
    non0    : (V, Z) -> Union(REC, "failed")
    nonzero?: V -> Boolean
    mkMat   : (M, List Z) -> M
    diagSubMatrix: M -> Union(Record(val:R, mat:M), "failed")
    determinantOfMinor: M -> R
    enumerateBinomial: (List Z, Z, Z) -> List Z

    nonzero? v == any?(#1 ~= 0, v)

-- returns [a, i, rown] if v = [0,...,0,a,0,...,0]
-- where a <> 0 and i is the index of a, "failed" otherwise.
    non0(v, rown) ==
      ans:REC
      allZero:Boolean := true
      for i in minIndex v .. maxIndex v repeat
        if qelt(v, i) ~= 0 then
          if allZero then
            allZero := false
            ans := [qelt(v, i), i, rown]
          else return "failed"
      allZero => "failed"
      ans

-- returns a matrix made from the non-zero rows of x whose row number
-- is not in l
    mkMat(x, l) ==
      empty?(ll := [parts row(x, i)
         for i in minRowIndex x .. maxRowIndex x |
           (not member?(i, l)) and nonzero? row(x, i)]$List(List R)) =>
              zero(1, ncols x)
      matrix ll

-- returns [m, d] where m = x with the zero rows and the rows of
-- the diagonal of d removed, if x has a diagonal submatrix of d's,
-- "failed" otherwise.
    diagSubMatrix x ==
      l  := [u::REC for i in minRowIndex x .. maxRowIndex x |
                                     (u := non0(row(x, i), i)) case REC]
      for a in removeDuplicates([r.val for r in l]$List(R)) repeat
        {[r.cl for r in l | r.val = a]$List(Z)}$Set(Z) =
          {[z for z in minColIndex x .. maxColIndex x]$List(Z)}$Set(Z)
            => return [a, mkMat(x, [r.rw for r in l | a = r.val])]
      "failed"

-- returns a non-zero determinant of a minor of x of rank equal to
-- the number of columns of x, if there is one, 0 otherwise
    determinantOfMinor x ==
-- do not compute a modulus for square matrices, since this is as expensive
-- as the Hermite reduction itself
      (nr := nrows x) <= (nc := ncols x) => 0
      lc := [i for i in minColIndex x .. maxColIndex x]$List(Integer)
      lr := [i for i in minRowIndex x .. maxRowIndex x]$List(Integer)
      for i in 1..(n := binomial(nr, nc)) repeat
        (d := determinant x(enumerateBinomial(lr, nc, i), lc)) ~= 0 =>
          j := i + 1 + (random()$Z rem (n - i))
          return gcd(d, determinant x(enumerateBinomial(lr, nc, j), lc))
      0

-- returns the i-th selection of m elements of l = (a1,...,an),
--                 /n\
-- where 1 <= i <= | |
--                 \m/
    enumerateBinomial(l, m, i) ==
      m1 := minIndex l - 1
      zero?(m := m - 1) => [l(m1 + i)]
      for j in 1..(n := #l) repeat
        i <= (b := binomial(n - j, m)) =>
          return concat(l(m1 + j), enumerateBinomial(rest(l, j), m, i))
        i := i - b
      error "Should not happen"

    rowEch x ==
      (u := diagSubMatrix x) case "failed" =>
        zero?(d := determinantOfMinor x) => rowEchelon x
        rowEchelon(x, d)
      rowEchelon(u.mat, u.val)

    vconc(y, m) ==
      vertConcat(diagonalMatrix new(ncols y, m)$V, map(#1 rem m, y))

    order(m, p) ==
      zero? m => -1
      for i in 0.. repeat
        (mm := m exquo p) case "failed" => return i
        m := mm::R

    if R has IntegerNumberSystem then
        normalizedDivide(n:R, d:R):Record(quotient:R, remainder:R) ==
            qr := divide(n, d)
            qr.remainder >= 0 => qr
            positive? d =>
                qr.remainder := qr.remainder + d
                qr.quotient := qr.quotient - 1
                qr
            qr.remainder := qr.remainder - d
            qr.quotient := qr.quotient + 1
            qr
    else
        normalizedDivide(n:R, d:R):Record(quotient:R, remainder:R) ==
            divide(n, d)

    rowEchLocal(x,p) ==
      (u := diagSubMatrix x) case "failed" =>
        zero?(d := determinantOfMinor x) => rowEchelon x
        rowEchelonLocal(x, d, p)
      rowEchelonLocal(u.mat, u.val, p)

    rowEchelonLocal(y, m, p) ==
        m := p**(order(m,p)::NonNegativeInteger)
        x     := vconc(y, m)
        nrows := maxRowIndex x
        ncols := maxColIndex x
        minr  := i := minRowIndex x
        for j in minColIndex x .. ncols repeat
          if i > nrows then leave x
          rown := minr - 1
          pivord : Integer
          npivord : Integer
          for k in i .. nrows repeat
            qelt(x,k,j) = 0 => "next k"
            npivord := order(qelt(x,k,j),p)
            (rown = minr - 1) or (npivord  <  pivord) =>
                    rown := k
                    pivord := npivord
          rown = minr - 1 => "enuf"
          x := swapRows!(x, i, rown)
          (a, b, d) := extendedEuclidean(qelt(x,i,j), m)
          qsetelt!(x,i,j,d)
          pivot := d
          for k in j+1 .. ncols repeat
            qsetelt!(x,i,k, a * qelt(x,i,k) rem m)
          for k in i+1 .. nrows repeat
            zero? qelt(x,k,j) => "next k"
            q := (qelt(x,k,j) exquo pivot) :: R
            for k1 in j+1 .. ncols repeat
              v2 := (qelt(x,k,k1) - q * qelt(x,i,k1)) rem m
              qsetelt!(x, k, k1, v2)
            qsetelt!(x, k, j, 0)
          for k in minr .. i-1 repeat
            zero? qelt(x,k,j) => "enuf"
            qr    := normalizedDivide(qelt(x,k,j), pivot)
            qsetelt!(x,k,j, qr.remainder)
            for k1 in j+1 .. ncols x repeat
              qsetelt!(x,k,k1,
                     (qelt(x,k,k1) - qr.quotient * qelt(x,i,k1)) rem m)
          i := i+1
        x

    if R has Field then
      rowEchelon(y, m) == rowEchelon vconc(y, m)

    else

      rowEchelon(y, m) ==
        x     := vconc(y, m)
        nrows := maxRowIndex x
        ncols := maxColIndex x
        minr  := i := minRowIndex x
        for j in minColIndex x .. ncols repeat
          if i > nrows then leave
          rown := minr - 1
          for k in i .. nrows repeat
            if (qelt(x,k,j) ~= 0) and ((rown = minr - 1) or
                  sizeLess?(qelt(x,k,j), qelt(x,rown,j))) then rown := k
          rown = minr - 1 => "next j"
          x := swapRows!(x, i, rown)
          for k in i+1 .. nrows repeat
            zero? qelt(x,k,j) => "next k"
            (a, b, d) := extendedEuclidean(qelt(x,i,j), qelt(x,k,j))
            (b1, a1) :=
               ((qelt(x,i,j) exquo d)::R, (qelt(x,k,j) exquo d)::R)
            -- a*b1+a1*b = 1
            for k1 in j+1 .. ncols repeat
              v1 := (a  * qelt(x,i,k1) +  b * qelt(x,k,k1)) rem m
              v2 := (b1 * qelt(x,k,k1) - a1 * qelt(x,i,k1)) rem m
              qsetelt!(x, i, k1, v1)
              qsetelt!(x, k, k1, v2)
            qsetelt!(x, i, j, d)
            qsetelt!(x, k, j, 0)
          un := unitNormal qelt(x,i,j)
          qsetelt!(x,i,j,un.canonical)
          if un.associate ~= 1 then for jj in (j+1)..ncols repeat
              qsetelt!(x,i,jj,un.associate * qelt(x,i,jj))

          xij := qelt(x,i,j)
          for k in minr .. i-1 repeat
            zero? qelt(x,k,j) => "next k"
            qr    := normalizedDivide(qelt(x,k,j), xij)
            qsetelt!(x,k,j, qr.remainder)
            for k1 in j+1 .. ncols x repeat
              qsetelt!(x,k,k1,
                     (qelt(x,k,k1) - qr.quotient * qelt(x,i,k1)) rem m)
          i := i+1
        x

@
\section{domain FRMOD FramedModule}
<<domain FRMOD FramedModule>>=
)abbrev domain FRMOD FramedModule
++ Author: Manuel Bronstein
++ Date Created: 27 Jan 1989
++ Date Last Updated: 24 Jul 1990
++ Keywords: ideal, algebra, module.
++ Examples: )r FRIDEAL INPUT
++ Description: Module representation of fractional ideals.
FramedModule(R, F, UP, A, ibasis): Exports == Implementation where
  R     : EuclideanDomain
  F     : QuotientFieldCategory R
  UP    : UnivariatePolynomialCategory F
  A     : FramedAlgebra(F, UP)
  ibasis: Vector A

  VR  ==> Vector R
  VF  ==> Vector F
  VA  ==> Vector A
  M   ==> Matrix F

  Exports ==> Monoid with
    basis : %  -> VA
      ++ basis((f1,...,fn)) = the vector \spad{[f1,...,fn]}.
    norm  : %  -> F
      ++ norm(f) returns the norm of the module f.
    module: VA -> %
      ++ module([f1,...,fn]) = the module generated by \spad{(f1,...,fn)}
      ++ over R.
    if A has RetractableTo F then
      module: FractionalIdeal(R, F, UP, A) -> %
        ++ module(I) returns I viewed has a module over R.

  Implementation ==> add
    import MatrixCommonDenominator(R, F)
    import ModularHermitianRowReduction(R)

    Rep  := VA

    iflag?:Reference(Boolean) := ref true
    wflag?:Reference(Boolean) := ref true
    imat := new(#ibasis, #ibasis, 0)$M
    wmat := new(#ibasis, #ibasis, 0)$M

    rowdiv      : (VR, R)  -> VF
    vectProd    : (VA, VA) -> VA
    wmatrix     : VA -> M
    W2A         : VF -> A
    intmat      : () -> M
    invintmat   : () -> M
    getintmat   : () -> Boolean
    getinvintmat: () -> Boolean

    1                      == ibasis
    module(v:VA)           == v
    basis m                == m pretend VA
    rowdiv(r, f)           == [r.i / f for i in minIndex r..maxIndex r]
    coerce(m:%):OutputForm == coerce(basis m)$VA
    W2A v                  == represents(v * intmat())
    wmatrix v              == coordinates(v) * invintmat()

    getinvintmat() ==
      m := inverse(intmat())::M
      for i in minRowIndex m .. maxRowIndex m repeat
        for j in minColIndex m .. maxColIndex m repeat
          imat(i, j) := qelt(m, i, j)
      false

    getintmat() ==
      m := coordinates ibasis
      for i in minRowIndex m .. maxRowIndex m repeat
        for j in minColIndex m .. maxColIndex m repeat
          wmat(i, j) := qelt(m, i, j)
      false

    invintmat() ==
      if iflag?() then iflag?() := getinvintmat()
      imat

    intmat() ==
      if wflag?() then wflag?() := getintmat()
      wmat

    vectProd(v1, v2) ==
      k := minIndex(v := new(#v1 * #v2, 0)$VA)
      for i in minIndex v1 .. maxIndex v1 repeat
        for j in minIndex v2 .. maxIndex v2 repeat
          qsetelt!(v, k, qelt(v1, i) * qelt(v2, j))
          k := k + 1
      v pretend VA

    norm m ==
      #(basis m) ~= #ibasis => error "Module not of rank n"
      determinant(coordinates(basis m) * invintmat())

    m1 * m2 ==
      m := rowEch((cd := splitDenominator wmatrix(
                                     vectProd(basis m1, basis m2))).num)
      module [u for i in minRowIndex m .. maxRowIndex m |
                           (u := W2A rowdiv(row(m, i), cd.den)) ~= 0]$VA

    if A has RetractableTo F then
      module(i:FractionalIdeal(R, F, UP, A)) ==
        module(basis i) * module(ibasis)

@
\section{category FDIVCAT FiniteDivisorCategory}
<<category FDIVCAT FiniteDivisorCategory>>=
)abbrev category FDIVCAT FiniteDivisorCategory
++ Category for finite rational divisors on a curve
++ Author: Manuel Bronstein
++ Date Created: 19 May 1993
++ Date Last Updated: 19 May 1993
++ Description:
++ This category describes finite rational divisors on a curve, that
++ is finite formal sums SUM(n * P) where the n's are integers and the
++ P's are finite rational points on the curve.
++ Keywords: divisor, algebraic, curve.
++ Examples: )r FDIV INPUT
FiniteDivisorCategory(F, UP, UPUP, R): Category == Result where
  F   : Field
  UP  : UnivariatePolynomialCategory F
  UPUP: UnivariatePolynomialCategory Fraction UP
  R   : FunctionFieldCategory(F, UP, UPUP)

  ID  ==> FractionalIdeal(UP, Fraction UP, UPUP, R)

  Result ==> AbelianGroup with
    ideal      : % -> ID
      ++ ideal(D) returns the ideal corresponding to a divisor D.
    divisor    : ID -> %
      ++ divisor(I) makes a divisor D from an ideal I.
    divisor    : R -> %
      ++ divisor(g) returns the divisor of the function g.
    divisor    : (F, F) -> %
      ++ divisor(a, b) makes the divisor P: \spad{(x = a, y = b)}.
      ++ Error: if P is singular.
    divisor    : (F, F, Integer) -> %
      ++ divisor(a, b, n) makes the divisor
      ++ \spad{nP} where P: \spad{(x = a, y = b)}.
      ++ P is allowed to be singular if n is a multiple of the rank.
    decompose  : % -> Record(id:ID, principalPart: R)
      ++ decompose(d) returns \spad{[id, f]} where \spad{d = (id) + div(f)}.
    reduce     : % -> %
      ++ reduce(D) converts D to some reduced form (the reduced forms can
      ++ be differents in different implementations).
    principal? : % -> Boolean
      ++ principal?(D) tests if the argument is the divisor of a function.
    generator  : % -> Union(R, "failed")
      ++ generator(d) returns f if \spad{(f) = d},
      ++ "failed" if d is not principal.
    divisor    : (R, UP, UP, UP, F) -> %
      ++ divisor(h, d, d', g, r) returns the sum of all the finite points
      ++ where \spad{h/d} has residue \spad{r}.
      ++ \spad{h} must be integral.
      ++ \spad{d} must be squarefree.
      ++ \spad{d'} is some derivative of \spad{d} (not necessarily dd/dx).
      ++ \spad{g = gcd(d,discriminant)} contains the ramified zeros of \spad{d}
   add
    principal? d == generator(d) case R

@
\section{domain HELLFDIV HyperellipticFiniteDivisor}
<<domain HELLFDIV HyperellipticFiniteDivisor>>=
)abbrev domain HELLFDIV HyperellipticFiniteDivisor
++ Finite rational divisors on an hyperelliptic curve
++ Author: Manuel Bronstein
++ Date Created: 19 May 1993
++ Date Last Updated: 20 July 1998
++ Description:
++ This domains implements finite rational divisors on an hyperelliptic curve,
++ that is finite formal sums SUM(n * P) where the n's are integers and the
++ P's are finite rational points on the curve.
++ The equation of the curve must be  y^2 = f(x) and f must have odd degree.
++ Keywords: divisor, algebraic, curve.
++ Examples: )r FDIV INPUT
HyperellipticFiniteDivisor(F, UP, UPUP, R): Exports == Implementation where
  F   : Field
  UP  : UnivariatePolynomialCategory F
  UPUP: UnivariatePolynomialCategory Fraction UP
  R   : FunctionFieldCategory(F, UP, UPUP)

  O   ==> OutputForm
  Z   ==> Integer
  RF  ==> Fraction UP
  ID  ==> FractionalIdeal(UP, RF, UPUP, R)
  ERR ==> error "divisor: incomplete implementation for hyperelliptic curves"

  Exports ==> FiniteDivisorCategory(F, UP, UPUP, R)

  Implementation ==> add
    if (uhyper:Union(UP, "failed") := hyperelliptic()$R) case "failed" then
              error "HyperellipticFiniteDivisor: curve must be hyperelliptic"

-- we use the semi-reduced representation from D.Cantor, "Computing in the
-- Jacobian of a HyperellipticCurve", Mathematics of Computation, vol 48,
-- no.177, January 1987, 95-101.
-- The representation [a,b,f] for D means D = [a,b] + div(f)
-- and [a,b] is a semi-reduced representative on the Jacobian
    Rep := Record(center:UP, polyPart:UP, principalPart:R, reduced?:Boolean)

    hyper:UP := uhyper::UP
    gen:Z    := ((degree(hyper)::Z - 1) exquo 2)::Z     -- genus of the curve
    dvd:O    := 'div::O
    zer:O    := 0::Z::O

    makeDivisor  : (UP, UP, R) -> %
    intReduc     : (R, UP) -> R
    princ?       : % -> Boolean
    polyIfCan    : R -> Union(UP, "failed")
    redpolyIfCan : (R, UP) -> Union(UP, "failed")
    intReduce    : (R, UP) -> R
    mkIdeal      : (UP, UP) -> ID
    reducedTimes : (Z, UP, UP) -> %
    reducedDouble: (UP, UP) -> %

    0                    == divisor(1$R)
    divisor(g:R)         == [1, 0, g, true]
    makeDivisor(a, b, g) == [a, b, g, false]
    princ? d             == one?(d.center) and zero?(d.polyPart)
    ideal d     == ideal([d.principalPart]) * mkIdeal(d.center, d.polyPart)
    decompose d == [ideal makeDivisor(d.center, d.polyPart, 1), d.principalPart]
    mkIdeal(a, b) == ideal [a::RF::R, reduce(monomial(1, 1)$UPUP - b::RF::UPUP)]

-- keep the sum reduced if d1 and d2 are both reduced at the start
    d1 + d2 ==
      a1  := d1.center;   a2 := d2.center
      b1  := d1.polyPart; b2 := d2.polyPart
      rec := principalIdeal [a1, a2, b1 + b2]
      d   := rec.generator
      h   := rec.coef              -- d = h1 a1 + h2 a2 + h3(b1 + b2)
      a   := ((a1 * a2) exquo d**2)::UP
      b:UP:= first(h) * a1 * b2
      b   := b + second(h) * a2 * b1
      b   := b + third(h) * (b1*b2 + hyper)
      b   := (b exquo d)::UP rem a
      dd  := makeDivisor(a, b, d::RF * d1.principalPart * d2.principalPart)
      d1.reduced? and d2.reduced? => reduce dd
      dd

-- if is cheaper to keep on reducing as we exponentiate if d is already reduced
    n:Z * d:% ==
      zero? n => 0
      negative? n => (-n) * (-d)
      divisor(d.principalPart ** n) + divisor(mkIdeal(d.center,d.polyPart) ** n)

    divisor(i:ID) ==
      one?(n := #(v := basis minimize i)) => divisor v minIndex v
      n ~= 2 => ERR
      a := v minIndex v
      h := v maxIndex v
      (u := polyIfCan a) case UP =>
        (w := redpolyIfCan(h, u::UP)) case UP => makeDivisor(u::UP, w::UP, 1)
        ERR
      (u := polyIfCan h) case UP =>
        (w := redpolyIfCan(a, u::UP)) case UP => makeDivisor(u::UP, w::UP, 1)
        ERR
      ERR

    polyIfCan a ==
      (u := retractIfCan(a)@Union(RF, "failed")) case "failed" => "failed"
      (v := retractIfCan(u::RF)@Union(UP, "failed")) case "failed" => "failed"
      v::UP

    redpolyIfCan(h, a) ==
      degree(p := lift h) ~= 1 => "failed"
      q := - coefficient(p, 0) / coefficient(p, 1)
      rec := extendedEuclidean(denom q, a)
      not ground?(rec.generator) => "failed"
      ((numer(q) * rec.coef1) exquo rec.generator)::UP rem a

    coerce(d:%):O ==
      r := bracket [d.center::O, d.polyPart::O]
      g := prefix(dvd, [d.principalPart::O])
      z := one?(d.principalPart)
      princ? d => (z => zer; g)
      z => r
      r + g

    reduce d ==
      d.reduced? => d
      degree(a := d.center) <= gen => (d.reduced? := true; d)
      b  := d.polyPart
      a0 := ((hyper - b**2) exquo a)::UP
      b0 := (-b) rem a0
      g  := d.principalPart * reduce(b::RF::UPUP-monomial(1,1)$UPUP) / a0::RF::R
      reduce makeDivisor(a0, b0, g)

    generator d ==
      d := reduce d
      princ? d => d.principalPart
      "failed"

    - d ==
      a := d.center
      makeDivisor(a, - d.polyPart, inv(a::RF * d.principalPart))

    d1 = d2 ==
      d1 := reduce d1
      d2 := reduce d2
      d1.center = d2.center and d1.polyPart = d2.polyPart
        and d1.principalPart = d2.principalPart

    divisor(a, b) ==
      x := monomial(1, 1)$UP
      not ground? gcd(d := x - a::UP, retract(discriminant())@UP) =>
                                  error "divisor: point is singular"
      makeDivisor(d, b::UP, 1)

    intReduce(h, b) ==
      v := integralCoordinates(h).num
      integralRepresents(
                [qelt(v, i) rem b for i in minIndex v .. maxIndex v], 1)

-- with hyperelliptic curves, it is cheaper to keep divisors in reduced form
    divisor(h, a, dp, g, r) ==
      h  := h - (r * dp)::RF::R
      a  := gcd(a, retract(norm h)@UP)
      h  := intReduce(h, a)
      if not ground? gcd(g, a) then h := intReduce(h ** rank(), a)
      hh := lift h
      b  := - coefficient(hh, 0) / coefficient(hh, 1)
      rec := extendedEuclidean(denom b, a)
      not ground?(rec.generator) => ERR
      bb := ((numer(b) * rec.coef1) exquo rec.generator)::UP rem a
      reduce makeDivisor(a, bb, 1)

@

\section{domain FDIV FiniteDivisor}

<<domain FDIV FiniteDivisor>>=
import Vector
)abbrev domain FDIV FiniteDivisor
++ Finite rational divisors on a curve
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Date Last Updated: 29 July 1993
++ Description:
++ This domains implements finite rational divisors on a curve, that
++ is finite formal sums SUM(n * P) where the n's are integers and the
++ P's are finite rational points on the curve.
++ Keywords: divisor, algebraic, curve.
++ Examples: )r FDIV INPUT
FiniteDivisor(F, UP, UPUP, R): Exports == Implementation where
  F   : Field
  UP  : UnivariatePolynomialCategory F
  UPUP: UnivariatePolynomialCategory Fraction UP
  R   : FunctionFieldCategory(F, UP, UPUP)

  N   ==> NonNegativeInteger
  RF  ==> Fraction UP
  ID  ==> FractionalIdeal(UP, RF, UPUP, R)

  Exports ==> FiniteDivisorCategory(F, UP, UPUP, R) with
    finiteBasis: % -> Vector R
      ++ finiteBasis(d) returns a basis for d as a module over {\em K[x]}.
    lSpaceBasis: % -> Vector R
      ++ lSpaceBasis(d) returns a basis for \spad{L(d) = {f | (f) >= -d}}
      ++ as a module over \spad{K[x]}.

  Implementation ==> add
    if hyperelliptic()$R case UP then
      Rep := HyperellipticFiniteDivisor(F, UP, UPUP, R)

      0                       == 0$Rep
      coerce(d:$):OutputForm  == coerce(d)$Rep
      d1 = d2                 == d1 =$Rep d2
      n:Integer * d:%         == n *$Rep d
      d1 + d2                 == d1 +$Rep d2
      - d                     == -$Rep d
      ideal d                 == ideal(d)$Rep
      reduce d                == reduce(d)$Rep
      generator d             == generator(d)$Rep
      decompose d             == decompose(d)$Rep
      divisor(i:ID)           == divisor(i)$Rep
      divisor(f:R)            == divisor(f)$Rep
      divisor(a, b)           == divisor(a, b)$Rep
      divisor(a, b, n)        == divisor(a, b, n)$Rep
      divisor(h, d, dp, g, r) == divisor(h, d, dp, g, r)$Rep

    else
      Rep := Record(id:ID, fbasis:Vector(R))

      import CommonDenominator(UP, RF, Vector RF)
      import UnivariatePolynomialCommonDenominator(UP, RF, UPUP)

      makeDivisor : (UP, UPUP, UP) -> %
      intReduce   : (R, UP) -> R

      ww := integralBasis()$R

      0                       == [1, empty()]
      divisor(i:ID)           == [i, empty()]
      divisor(f:R)            == divisor ideal [f]
      coerce(d:%):OutputForm  == ideal(d)::OutputForm
      ideal d                 == d.id
      decompose d             == [ideal d, 1]
      d1 = d2                 == basis(ideal d1) = basis(ideal d2)
      n: Integer * d:%        == divisor(ideal(d) ** n)
      d1 + d2                 == divisor(ideal d1 * ideal d2)
      - d                     == divisor inv ideal d
      divisor(h, d, dp, g, r) == makeDivisor(d, lift h - (r * dp)::RF::UPUP, g)

      intReduce(h, b) ==
        v := integralCoordinates(h).num
        integralRepresents(
                      [qelt(v, i) rem b for i in minIndex v .. maxIndex v], 1)

      divisor(a, b) ==
        x := monomial(1, 1)$UP
        not ground? gcd(d := x - a::UP, retract(discriminant())@UP) =>
                                          error "divisor: point is singular"
        makeDivisor(d, monomial(1, 1)$UPUP - b::UP::RF::UPUP, 1)

      divisor(a, b, n) ==
        not(ground? gcd(d := monomial(1, 1)$UP - a::UP,
            retract(discriminant())@UP)) and
                  ((n exquo rank()) case "failed") =>
                                    error "divisor: point is singular"
        m:N :=
          negative? n => (-n)::N
          n::N
        g := makeDivisor(d**m,(monomial(1,1)$UPUP - b::UP::RF::UPUP)**m,1)
        negative? n => -g
        g

      reduce d ==
        (i := minimize(j := ideal d)) = j => d
        #(n := numer i) ~= 2 => divisor i
        cd := splitDenominator lift n(1 + minIndex n)
        b  := gcd(cd.den * retract(retract(n minIndex n)@RF)@UP,
                  retract(norm reduce(cd.num))@UP)
        e  := cd.den * denom i
        divisor ideal([(b / e)::R,
                reduce map((retract(#1)@UP rem b) / e, cd.num)]$Vector(R))

      finiteBasis d ==
        if empty?(d.fbasis) then
          d.fbasis := normalizeAtInfinity
                        basis module(ideal d)$FramedModule(UP, RF, UPUP, R, ww)
        d.fbasis

      generator d ==
        bsis := finiteBasis d
        for i in minIndex bsis .. maxIndex bsis repeat
          integralAtInfinity? qelt(bsis, i) =>
            return primitivePart qelt(bsis,i)
        "failed"

      lSpaceBasis d ==
        map!(primitivePart, reduceBasisAtInfinity finiteBasis(-d))

-- b = center, hh = integral function, g = gcd(b, discriminant)
      makeDivisor(b, hh, g) ==
        b := gcd(b, retract(norm(h := reduce hh))@UP)
        h := intReduce(h, b)
        if not ground? gcd(g, b) then h := intReduce(h ** rank(), b)
        divisor ideal [b::RF::R, h]$Vector(R)

@
\section{package FDIV2 FiniteDivisorFunctions2}
<<package FDIV2 FiniteDivisorFunctions2>>=
)abbrev package FDIV2 FiniteDivisorFunctions2
++ Lift a map to finite divisors.
++ Author: Manuel Bronstein
++ Date Created: 1988
++ Date Last Updated: 19 May 1993
FiniteDivisorFunctions2(R1, UP1, UPUP1, F1, R2, UP2, UPUP2, F2):
 Exports == Implementation where
  R1   : Field
  UP1  : UnivariatePolynomialCategory R1
  UPUP1: UnivariatePolynomialCategory Fraction UP1
  F1   : FunctionFieldCategory(R1, UP1, UPUP1)
  R2   : Field
  UP2  : UnivariatePolynomialCategory R2
  UPUP2: UnivariatePolynomialCategory Fraction UP2
  F2   : FunctionFieldCategory(R2, UP2, UPUP2)

  Exports ==> with
    map: (R1 -> R2, FiniteDivisor(R1, UP1, UPUP1, F1)) ->
                                       FiniteDivisor(R2, UP2, UPUP2, F2)
	++ map(f,d) \undocumented{} 

  Implementation ==> add
    import UnivariatePolynomialCategoryFunctions2(R1,UP1,R2,UP2)
    import FunctionFieldCategoryFunctions2(R1,UP1,UPUP1,F1,R2,UP2,UPUP2,F2)
    import FractionalIdealFunctions2(UP1, Fraction UP1, UPUP1, F1,
                                     UP2, Fraction UP2, UPUP2, F2)

    map(f, d) ==
      rec := decompose d
      divisor map(f, rec.principalPart) + divisor map(map(f, #1), rec.id)

@
\section{License}
<<license>>=
--Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd.
--All rights reserved.
--
--Redistribution and use in source and binary forms, with or without
--modification, are permitted provided that the following conditions are
--met:
--
--    - Redistributions of source code must retain the above copyright
--      notice, this list of conditions and the following disclaimer.
--
--    - Redistributions in binary form must reproduce the above copyright
--      notice, this list of conditions and the following disclaimer in
--      the documentation and/or other materials provided with the
--      distribution.
--
--    - Neither the name of The Numerical ALgorithms Group Ltd. nor the
--      names of its contributors may be used to endorse or promote products
--      derived from this software without specific prior written permission.
--
--THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS
--IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED
--TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
--PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER
--OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL,
--EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
--PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR
--PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF
--LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING
--NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
--SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
@
<<*>>=
<<license>>

-- SPAD files for the algebraic integration world should be compiled
-- in the following order:
--
--   curve DIVISOR reduc pfo intalg int

<<domain FRIDEAL FractionalIdeal>>
<<package FRIDEAL2 FractionalIdealFunctions2>>
<<package MHROWRED ModularHermitianRowReduction>>
<<domain FRMOD FramedModule>>
<<category FDIVCAT FiniteDivisorCategory>>
<<domain HELLFDIV HyperellipticFiniteDivisor>>
<<domain FDIV FiniteDivisor>>
<<package FDIV2 FiniteDivisorFunctions2>>
@
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}