\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} <>= )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::$) i @ \section{package FRIDEAL2 FractionalIdealFunctions2} <>= )abbrev package FRIDEAL2 FractionalIdealFunctions2 ++ Lifting of morphisms to fractional ideals. ++ Author: Manuel Bronstein ++ Date Created: 1 Feb 1989 ++ Date Last Updated: 27 Feb 1990 ++ Keywords: ideal, algebra, module. FractionalIdealFunctions2(R1, F1, U1, A1, R2, F2, U2, A2): Exports == Implementation where R1, R2: EuclideanDomain F1: QuotientFieldCategory R1 U1: UnivariatePolynomialCategory F1 A1: Join(FramedAlgebra(F1, U1), RetractableTo F1) F2: QuotientFieldCategory R2 U2: UnivariatePolynomialCategory F2 A2: Join(FramedAlgebra(F2, U2), RetractableTo F2) Exports ==> with map: (R1 -> R2, FractionalIdeal(R1, F1, U1, A1)) -> FractionalIdeal(R2, F2, U2, A2) ++ map(f,i) \undocumented{} Implementation ==> add fmap: (F1 -> F2, A1) -> A2 fmap(f, a) == v := coordinates a represents [f qelt(v, i) for i in minIndex v .. maxIndex v]$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} <>= )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} <>= )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} <>= )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} <>= )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)$UP not ground? gcd(d := x - a::UP, retract(discriminant())@UP) => error "divisor: point is singular" makeDivisor(d, b::UP, 1) intReduce(h, b) == v := integralCoordinates(h).num integralRepresents( [qelt(v, i) rem b for i in minIndex v .. maxIndex v], 1) -- with hyperelliptic curves, it is cheaper to keep divisors in reduced form divisor(h, a, dp, g, r) == h := h - (r * dp)::RF::R a := gcd(a, retract(norm h)@UP) h := intReduce(h, a) if not ground? gcd(g, a) then h := intReduce(h ** rank(), a) hh := lift h b := - coefficient(hh, 0) / coefficient(hh, 1) rec := extendedEuclidean(denom b, a) not ground?(rec.generator) => ERR bb := ((numer(b) * rec.coef1) exquo rec.generator)::UP rem a reduce makeDivisor(a, bb, 1) @ \section{domain FDIV FiniteDivisor} <>= import Vector )abbrev domain FDIV FiniteDivisor ++ Finite rational divisors on a curve ++ Author: Manuel Bronstein ++ Date Created: 1987 ++ Date Last Updated: 29 July 1993 ++ Description: ++ This domains implements finite rational divisors on a curve, that ++ is finite formal sums SUM(n * P) where the n's are integers and the ++ P's are finite rational points on the curve. ++ Keywords: divisor, algebraic, curve. ++ Examples: )r FDIV INPUT FiniteDivisor(F, UP, UPUP, R): Exports == Implementation where F : Field UP : UnivariatePolynomialCategory F UPUP: UnivariatePolynomialCategory Fraction UP R : FunctionFieldCategory(F, UP, UPUP) N ==> NonNegativeInteger RF ==> Fraction UP ID ==> FractionalIdeal(UP, RF, UPUP, R) Exports ==> FiniteDivisorCategory(F, UP, UPUP, R) with finiteBasis: % -> Vector R ++ finiteBasis(d) returns a basis for d as a module over {\em K[x]}. lSpaceBasis: % -> Vector R ++ lSpaceBasis(d) returns a basis for \spad{L(d) = {f | (f) >= -d}} ++ as a module over \spad{K[x]}. Implementation ==> add if hyperelliptic()$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} <>= )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} <>= --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. @ <<*>>= <> -- SPAD files for the algebraic integration world should be compiled -- in the following order: -- -- curve DIVISOR reduc pfo intalg int <> <> <> <> <> <> <> <> @ \eject \begin{thebibliography}{99} \bibitem{1} nothing \end{thebibliography} \end{document}