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dodane rozwiniecie w niesk

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jgarnek 2022-10-11 19:17:22 +00:00
parent f13f890270
commit d30ead5c7d
1 changed files with 269 additions and 35 deletions
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282
superelliptic.ipynb
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@ -2,7 +2,7 @@
"cells": [ "cells": [
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 54, "execution_count": 55,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
@ -271,8 +271,8 @@
" g2 = other.function\n", " g2 = other.function\n",
" g = reduction(C, g1 / g2)\n", " g = reduction(C, g1 / g2)\n",
" return superelliptic_function(C, g)\n", " return superelliptic_function(C, g)\n",
" \n", "\n",
"def diffn(self):\n", " def diffn(self):\n",
" C = self.curve\n", " C = self.curve\n",
" f = C.polynomial\n", " f = C.polynomial\n",
" m = C.exponent\n", " m = C.exponent\n",
@ -284,7 +284,53 @@
" A = g.derivative(x)\n", " A = g.derivative(x)\n",
" B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))\n", " B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))\n",
" return superelliptic_form(C, A+B)\n", " return superelliptic_form(C, A+B)\n",
"\n",
" \n", " \n",
" def expansion_at_infty(self, i = 0, prec=10):\n",
" C = self.curve\n",
" f = C.polynomial\n",
" m = C.exponent\n",
" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" f = Rx(f)\n",
" Rt.<t> = LaurentSeriesRing(F, default_prec=prec)\n",
" RptW.<W> = PolynomialRing(Rt)\n",
" RptWQ = FractionField(RptW)\n",
" Rxy.<x, y> = PolynomialRing(F)\n",
" RxyQ = FractionField(Rxy)\n",
" fct = self.function\n",
" fct = RxyQ(fct)\n",
" r = f.degree()\n",
" delta, a, b = xgcd(m, r)\n",
" b = -b\n",
" M = m/delta\n",
" R = r/delta\n",
" while a<0:\n",
" a += R\n",
" b += M\n",
" \n",
" g = (x^r*f(x = 1/x))\n",
" gW = RptWQ(g(x = t^M * W^b)) - W^(delta)\n",
" ww = naive_hensel(gW, F, start = root_of_unity(F, delta), prec = prec)\n",
" xx = Rt(1/(t^M*ww^b))\n",
" yy = 1/(t^R*ww^a)\n",
" return Rt(fct(x = Rt(xx), y = Rt(yy)))\n",
" \n",
"def naive_hensel(fct, F, start = 1, prec=10):\n",
" Rt.<t> = LaurentSeriesRing(F, default_prec=prec)\n",
" RtQ = FractionField(Rt)\n",
" RptW.<W> = PolynomialRing(RtQ)\n",
" fct = RptW(fct)\n",
" alpha = (fct.derivative())(W = start)\n",
" w0 = Rt(1)\n",
" i = 1\n",
" while(i < prec):\n",
" a = 0\n",
" while(Rt(fct(W = w0)).valuation()) <= i:\n",
" w0 = w0 - fct(W = w0)/alpha\n",
" i += 1\n",
" return w0\n",
"\n",
"class superelliptic_form:\n", "class superelliptic_form:\n",
" def __init__(self, C, g):\n", " def __init__(self, C, g):\n",
" F = C.base_ring\n", " F = C.base_ring\n",
@ -405,12 +451,20 @@
" if(-d*M + j*R -(M+1)<0):\n", " if(-d*M + j*R -(M+1)<0):\n",
" return 0\n", " return 0\n",
" return 1\n", " return 1\n",
" \n", "\n",
" def expansion_at_infty(self, i = 0, prec=10):\n",
" g = self.form\n",
" C = self.curve\n",
" g = superelliptic_function(C, g)\n",
" g = g.expansion_at_infty(i = i, prec=prec)\n",
" x_series = superelliptic_function(C, x).expansion_at_infty(i = i, prec=prec)\n",
" dx_series = x_series.derivative()\n",
" return g*dx_series\n",
" \n", " \n",
"class superelliptic_cech:\n", "class superelliptic_cech:\n",
" def __init__(self, C, omega, fct):\n", " def __init__(self, C, omega, fct):\n",
" self.omega0 = omega\n", " self.omega0 = omega\n",
" self.omega8 = omega - diffn(fct)\n", " self.omega8 = omega - fct.diffn()\n",
" self.f = fct\n", " self.f = fct\n",
" self.curve = C\n", " self.curve = C\n",
" \n", " \n",
@ -560,18 +614,17 @@
" if (i%p) == p-1:\n", " if (i%p) == p-1:\n",
" power = Integer((i-(p-1))/p)\n", " power = Integer((i-(p-1))/p)\n",
" result += Integer(h[i]) * x^(power) \n", " result += Integer(h[i]) * x^(power) \n",
" return result" " return result\n",
] "\n",
}, "#Find delta-th root of unity in field F\n",
{ "def root_of_unity(F, delta):\n",
"cell_type": "code", " Rx.<x> = PolynomialRing(F)\n",
"execution_count": 55, " cyclotomic = x^(delta) - 1\n",
"metadata": { " for root, a in cyclotomic.roots():\n",
"collapsed": false " powers = [root^d for d in delta.divisors() if d!= delta]\n",
}, " if 1 not in powers:\n",
"outputs": [ " return root\n",
], " \n",
"source": [
"def preimage(U, V, M): #preimage of subspace U under M\n", "def preimage(U, V, M): #preimage of subspace U under M\n",
" basis_preimage = M.right_kernel().basis()\n", " basis_preimage = M.right_kernel().basis()\n",
" imageU = U.intersection(M.transpose().image())\n", " imageU = U.intersection(M.transpose().image())\n",
@ -657,7 +710,18 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 58, "execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
@ -671,7 +735,19 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 59, "execution_count": 58,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"omega = C.de_rham_basis()[3]"
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": { "metadata": {
"collapsed": false "collapsed": false
}, },
@ -679,19 +755,176 @@
{ {
"data": { "data": {
"text/plain": [ "text/plain": [
"[0 0 1]\n", "3*t^2 + 3*t^14 + t^16 + 3*t^26 + 3*t^28 + 3*t^30 + 3*t^38 + t^40 + 3*t^50 + 3*t^62 + t^72 + t^74 + 2*t^76 + 4*t^84 + 2*t^86 + 4*t^88 + 4*t^90 + 2*t^96 + 4*t^98 + t^100 + O(t^114)"
"[0 2 0]\n",
"[1 2 0]"
] ]
}, },
"execution_count": 59, "execution_count": 51,
"metadata": { "metadata": {
}, },
"output_type": "execute_result" "output_type": "execute_result"
} }
], ],
"source": [ "source": [
"C.cartier_matrix()" "omega.omega8().exp"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"F = QQ\n",
"Rt.<t> = LaurentSeriesRing(F, default_prec=20)\n",
"RtQ = FractionField(Rt)\n",
"RptW.<W> = PolynomialRing(RtQ)\n",
"fct = W^2 - (1+t)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"A = naive_hensel(fct, F, start = 1, prec=10)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"WARNING: Some output was deleted.\n"
]
}
],
"source": [
"A^2 - (1+t)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"FF = GF(5^6)"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[(4, 1),\n",
" (1, 1),\n",
" (3*z6^5 + 4*z6^4 + 3*z6^2 + 2*z6 + 2, 1),\n",
" (3*z6^5 + 4*z6^4 + 3*z6^2 + 2*z6 + 1, 1),\n",
" (2*z6^5 + z6^4 + 2*z6^2 + 3*z6 + 4, 1),\n",
" (2*z6^5 + z6^4 + 2*z6^2 + 3*z6 + 3, 1)]"
]
},
"execution_count": 56,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Rx.<x> = PolynomialRing(FF)\n",
"(x^6 - 1).roots()"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"3*z2 + 3"
]
},
"execution_count": 57,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"root_of_unity(GF(5^2), 3)"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"x^2 + x + 1"
]
},
"execution_count": 52,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Rx.<x> = PolynomialRing(GF(5^3))\n",
"Rx(x^2 + x + 1).factor()"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"a = GF(5^3).random_element()"
] ]
}, },
{ {
@ -703,6 +936,7 @@
"outputs": [ "outputs": [
], ],
"source": [ "source": [
"a"
] ]
} }
], ],