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\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 members(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()$F * qelt(v, j) for j in minIndex v .. maxIndex v]
else
randomLC(m, v) ==
+/[random(m)$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 := [members 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]$List(Integer)
for i in 1..(n := binomial(nr, nc)) repeat
(d := determinant x(enumerateBinomial(lr, nc, i), lc)) ~= 0 =>
j := i + 1 + random(n - i)$Z
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 not one?(un.associate) 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 deref iflag? then setref(iflag?,getinvintmat())
imat
intmat() ==
if deref wflag? then setref(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)$UPUP - b::RF::UPUP)]
-- keep the sum reduced if d1 and d2 are both reduced at the start
d1 + d2 ==
a1 := d1.center; a2 := d2.center
b1 := d1.polyPart; b2 := d2.polyPart
rec := principalIdeal [a1, a2, b1 + b2]
d := rec.generator
h := rec.coef -- d = h1 a1 + h2 a2 + h3(b1 + b2)
a := ((a1 * a2) exquo d**2)::UP
b:UP:= first(h) * a1 * b2
b := b + second(h) * a2 * b1
b := b + third(h) * (b1*b2 + hyper)
b := (b exquo d)::UP rem a
dd := makeDivisor(a, b, d::RF * d1.principalPart * d2.principalPart)
d1.reduced? and d2.reduced? => reduce dd
dd
-- if is cheaper to keep on reducing as we exponentiate if d is already reduced
n:Z * d:% ==
zero? n => 0
negative? n => (-n) * (-d)
divisor(d.principalPart ** n) + divisor(mkIdeal(d.center,d.polyPart) ** n)
divisor(i:ID) ==
one?(n := #(v := basis minimize i)) => divisor v minIndex v
n ~= 2 => ERR
a := v minIndex v
h := v maxIndex v
(u := polyIfCan a) case UP =>
(w := redpolyIfCan(h, u::UP)) case UP => makeDivisor(u::UP, w::UP, 1)
ERR
(u := polyIfCan h) case UP =>
(w := redpolyIfCan(a, u::UP)) case UP => makeDivisor(u::UP, w::UP, 1)
ERR
ERR
polyIfCan a ==
(u := retractIfCan(a)@Union(RF, "failed")) case "failed" => "failed"
(v := retractIfCan(u::RF)@Union(UP, "failed")) case "failed" => "failed"
v::UP
redpolyIfCan(h, a) ==
not one? degree(p := lift h) => "failed"
q := - coefficient(p, 0) / coefficient(p, 1)
rec := extendedEuclidean(denom q, a)
not ground?(rec.generator) => "failed"
((numer(q) * rec.coef1) exquo rec.generator)::UP rem a
coerce(d:%):O ==
r := bracket [d.center::O, d.polyPart::O]
g := prefix(dvd, [d.principalPart::O])
z := one?(d.principalPart)
princ? d => (z => zer; g)
z => r
r + g
reduce d ==
d.reduced? => d
degree(a := d.center) <= gen => (d.reduced? := true; d)
b := d.polyPart
a0 := ((hyper - b**2) exquo a)::UP
b0 := (-b) rem a0
g := d.principalPart * reduce(b::RF::UPUP-monomial(1,1)$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}