/usr/share/axiom-20170501/src/algebra/RADFF.spad is in axiom-source 20170501-3.
This file is owned by root:root, with mode 0o644.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 | )abbrev domain RADFF RadicalFunctionField
++ Author: Manuel Bronstein
++ Date Created: 1987
++ Date Last Updated: 27 July 1993
++ Description:
++ Function field defined by y**n = f(x);
RadicalFunctionField(F, UP, UPUP, radicnd, n) : SIG == CODE where
F : UniqueFactorizationDomain
UP : UnivariatePolynomialCategory F
UPUP : UnivariatePolynomialCategory Fraction UP
radicnd : Fraction UP
n : NonNegativeInteger
N ==> NonNegativeInteger
Z ==> Integer
RF ==> Fraction UP
QF ==> Fraction UPUP
UP2 ==> SparseUnivariatePolynomial UP
REC ==> Record(factor:UP, exponent:Z)
MOD ==> monomial(1, n)$UPUP - radicnd::UPUP
INIT ==> if (deref brandNew?) then startUp false
SIG ==> FunctionFieldCategory(F, UP, UPUP)
CODE ==> SimpleAlgebraicExtension(RF, UPUP, MOD) add
import ChangeOfVariable(F, UP, UPUP)
import InnerCommonDenominator(UP, RF, Vector UP, Vector RF)
import UnivariatePolynomialCategoryFunctions2(RF, UPUP, UP, UP2)
diag : Vector RF -> Vector $
startUp : Boolean -> Void
fullVector : (Factored UP, N) -> PrimitiveArray UP
iBasis : (UP, N) -> Vector UP
inftyBasis : (RF, N) -> Vector RF
basisvec : () -> Vector RF
char0StartUp: () -> Void
charPStartUp: () -> Void
getInfBasis : () -> Void
radcand : () -> UP
charPintbas : (UPUP, RF, Vector RF, Vector RF) -> Void
brandNew?:Reference(Boolean) := ref true
discPoly:Reference(RF) := ref(0$RF)
newrad:Reference(UP) := ref(0$UP)
n1 := (n - 1)::N
modulus := MOD
ibasis:Vector(RF) := new(n, 0)
invibasis:Vector(RF) := new(n, 0)
infbasis:Vector(RF) := new(n, 0)
invinfbasis:Vector(RF):= new(n, 0)
mini := minIndex ibasis
discriminant() == (INIT; discPoly())
radcand() == (INIT; newrad())
integralBasis() == (INIT; diag ibasis)
integralBasisAtInfinity() == (INIT; diag infbasis)
basisvec() == (INIT; ibasis)
integralMatrix() == diagonalMatrix basisvec()
integralMatrixAtInfinity() == (INIT; diagonalMatrix infbasis)
inverseIntegralMatrix() == (INIT; diagonalMatrix invibasis)
inverseIntegralMatrixAtInfinity()==(INIT;diagonalMatrix invinfbasis)
definingPolynomial() == modulus
ramified?(point:F) == zero?(radcand() point)
branchPointAtInfinity?() == (degree(radcand()) exquo n) case "failed"
elliptic() == (n = 2 and degree(radcand()) = 3 => radcand(); "failed")
hyperelliptic() == (n=2 and odd? degree(radcand()) => radcand(); "failed")
diag v == [reduce monomial(qelt(v,i+mini), i) for i in 0..n1]
integralRepresents(v, d) ==
ib := basisvec()
represents
[qelt(ib, i) * (qelt(v, i) /$RF d) for i in mini .. maxIndex ib]
integralCoordinates f ==
v := coordinates f
ib := basisvec()
splitDenominator
[qelt(v,i) / qelt(ib,i) for i in mini .. maxIndex ib]$Vector(RF)
integralDerivationMatrix d ==
dlogp := differentiate(radicnd, d) / (n * radicnd)
v := basisvec()
cd := splitDenominator(
[(i - mini) * dlogp + differentiate(qelt(v, i), d) / qelt(v, i)
for i in mini..maxIndex v]$Vector(RF))
[diagonalMatrix(cd.num), cd.den]
-- return (d0,...,d(n-1)) s.t. (1/d0, y/d1,...,y**(n-1)/d(n-1))
-- is an integral basis for the curve y**d = p
-- requires that p has no factor of multiplicity >= d
iBasis(p, d) ==
pl := fullVector(squareFree p, d)
d1 := (d - 1)::N
[*/[pl.j ** ((i * j) quo d) for j in 0..d1] for i in 0..d1]
-- returns a vector [a0,a1,...,a_{m-1}] of length m such that
-- p = a0^0 a1^1 ... a_{m-1}^{m-1}
fullVector(p, m) ==
ans:PrimitiveArray(UP) := new(m, 0)
ans.0 := unit p
l := factors p
for i in 1..maxIndex ans repeat
ans.i :=
(u := find(s+->s.exponent = i, l)) case "failed" => 1
(u::REC).factor
ans
-- return (f0,...,f(n-1)) s.t. (f0, y f1,..., y**(n-1) f(n-1))
-- is a local integral basis at infinity for the curve y**d = p
inftyBasis(p, m) ==
rt := rootPoly(p(x := inv(monomial(1, 1)$UP :: RF)), m)
m ^= rt.exponent =>
error "Curve not irreducible after change of variable 0 -> infinity"
a := (rt.coef) x
b:RF := 1
v := iBasis(rt.radicand, m)
w:Vector(RF) := new(m, 0)
for i in mini..maxIndex v repeat
qsetelt_!(w, i, b / ((qelt(v, i)::RF) x))
b := b * a
w
charPintbas(p, c, v, w) ==
degree(p) ^= n => error "charPintbas: should not happen"
q:UP2 := map(s+->retract(s)@UP, p)
ib := integralBasis()$FunctionFieldIntegralBasis(UP, UP2,
SimpleAlgebraicExtension(UP, UP2, q))
not diagonal?(ib.basis)=>
error "charPintbas: integral basis not diagonal"
a:RF := 1
for i in minRowIndex(ib.basis) .. maxRowIndex(ib.basis)
for j in minColIndex(ib.basis) .. maxColIndex(ib.basis)
for k in mini .. maxIndex v repeat
qsetelt_!(v, k, (qelt(ib.basis, i, j) / ib.basisDen) * a)
qsetelt_!(w, k, qelt(ib.basisInv, i, j) * inv a)
a := a * c
void
charPStartUp() ==
r := mkIntegral modulus
charPintbas(r.poly, r.coef, ibasis, invibasis)
x := inv(monomial(1, 1)$UP :: RF)
invmod := monomial(1, n)$UPUP - (radicnd x)::UPUP
r := mkIntegral invmod
charPintbas(r.poly, (r.coef) x, infbasis, invinfbasis)
startUp b ==
brandNew?() := b
if zero?(p := characteristic()$F) or p > n then char0StartUp()
else charPStartUp()
dsc:RF := ((-1)$Z ** ((n *$N n1) quo 2::N) * (n::Z)**n)$Z *
radicnd ** n1 *
*/[qelt(ibasis, i) ** 2 for i in mini..maxIndex ibasis]
discPoly() := primitivePart(numer dsc) / denom(dsc)
void
char0StartUp() ==
rp := rootPoly(radicnd, n)
rp.exponent ^= n =>
error "RadicalFunctionField: curve is not irreducible"
newrad() := rp.radicand
ib := iBasis(newrad(), n)
infb := inftyBasis(radicnd, n)
invden:RF := 1
for i in mini..maxIndex ib repeat
qsetelt_!(invibasis, i, a := qelt(ib, i) * invden)
qsetelt_!(ibasis, i, inv a)
invden := invden / rp.coef -- always equals 1/rp.coef**(i-mini)
qsetelt_!(infbasis, i, a := qelt(infb, i))
qsetelt_!(invinfbasis, i, inv a)
void
ramified?(p:UP) ==
(r := retractIfCan(p)@Union(F, "failed")) case F =>
singular?(r::F)
(radcand() exquo p) case UP
singular?(p:UP) ==
(r := retractIfCan(p)@Union(F, "failed")) case F =>
singular?(r::F)
(radcand() exquo(p**2)) case UP
branchPoint?(p:UP) ==
(r := retractIfCan(p)@Union(F, "failed")) case F =>
branchPoint?(r::F)
((q := (radcand() exquo p)) case UP) and
((q::UP exquo p) case "failed")
singular?(point:F) ==
zero?(radcand() point) and
zero?(((radcand() exquo (monomial(1,1)$UP-point::UP))::UP) point)
branchPoint?(point:F) ==
zero?(radcand() point) and not
zero?(((radcand() exquo (monomial(1,1)$UP-point::UP))::UP) point)
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