321 lines
9.2 KiB
Plaintext
321 lines
9.2 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"id": "8565ca20-c37d-4b6c-8aa0-a32074c8e6e2",
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"metadata": {},
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"outputs": [],
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"source": [
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"# g = Graph(17)\n",
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"# a = graph_editor(g)\n",
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"# a.show()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "118f3510-30b1-4a0c-81e3-146325f5c8d4",
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"metadata": {},
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"outputs": [],
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"source": [
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"# A =g.adjacency_matrix()\n",
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"# A[0, 0] = -13\n",
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"# for i in (1, 3, 5, 6, 7):\n",
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"# A[i, i] = -2\n",
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"# A[i + 8, i + 8] = -2\n",
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"# for i in (2,):\n",
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"# A[i, i] = -4\n",
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"# A[i + 8, i + 8] = -4\n",
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"# for i in (4, 8):\n",
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"# A[i, i] = -1\n",
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"# A[i + 8, i + 8] = -1\n",
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"# # A = A.change_ring(QQ)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"id": "1c1485e6-9433-4b5b-93c4-20dc6d95926c",
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"metadata": {},
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"outputs": [],
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"source": [
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"A = Matrix(QQ, 17, [-13, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -4, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -4, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1])\n",
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"Vf = vector(4 * [0] + [1] + 7 * [0] + [1] + 4 * [0])\n",
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"Vg = vector(8 * [0] + [1] + 7 * [0] + [1])\n",
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"Vd = vector([13] + 7 * [0] + [1] + 7 * [0] + [1])\n",
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"R.<x, y, z> = QQ[]\n",
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"S.<s> = QQ[]\n",
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"T.<t> = QQ[]"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"id": "2d89bcf0-311e-4799-872c-72b29b23ee83",
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"metadata": {},
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"outputs": [],
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"source": [
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"def trinomiales(A, Vf, Vg, Vd, total=True):\n",
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" n = A.nrows()\n",
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" A1 = -A^-1\n",
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" Nf, Ng, Nd = [A1 * N0 for N0 in (Vf, Vg, Vd)]\n",
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" diag = vector(A.diagonal())\n",
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" unos = vector(n * [1])\n",
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" if total:\n",
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" VV = Vf + Vg\n",
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" else:\n",
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" VV = Vf.parent()(0)\n",
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" euler = (-A + 2 * identity_matrix(n)) * unos + diag - VV - Vd\n",
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" nus = A1 * (diag + 2 * unos) + unos\n",
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" expos = {}\n",
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" for j in range(n):\n",
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" a, b = (Nf[j], Ng[j])\n",
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" eq = 3 * Nf[j] == 2 * Ng[j]\n",
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" nnd = min(3 * Nf[j], 2 * Ng[j]) == Nd[j]\n",
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" if nnd:\n",
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" expos[j] = ((a, b), (a, b), nus[j], eq, nnd)\n",
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" else:\n",
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" expos[j] = ((a, b), (Nf[j], Nd[j] - Ng[j]), nus[j], eq, nnd)\n",
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" cadenas = ('d',)\n",
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" if total:\n",
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" expos['f'] = ((1, 0), (0, 0), 1, False, True)\n",
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" expos['g'] = ((0, 1), (0, 0), 1, False, True)\n",
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" cadenas = ('f', 'g') + cadenas\n",
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" expos['d'] = ((0, 0), (0, 1), 1, True, False)\n",
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" pols = {}\n",
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" for j in range(n):\n",
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" if expos[j][-1]:\n",
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" a, b = expos[j][0]\n",
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" pols[j] = (z^3 + x^a * z + x^b, (expos[j][2], 1, 1), euler[j])\n",
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" else:\n",
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" a, b = expos[j][1]\n",
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" pols[j] = (z^3 + x^a * z^2 + x^b, (expos[j][2], 1, 1), euler[j])\n",
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" interdiv = [(i, j) for i, j in Combinations(n, 2) if A[i, j] != 0]\n",
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" interstrict = [(i, j) for i in range(n) for j in cadenas if eval('V' + j)[i] != 0]\n",
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" for i,j in interdiv + interstrict:\n",
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" if j in ZZ:\n",
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" ep = 1\n",
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" else:\n",
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" ep = eval('V' + j)[i]\n",
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" eqi, nndi = expos[i][-2:]\n",
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" eqj, nndj = expos[j][-2:]\n",
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" nd = eqi and eqj and not (nndi and nndj)\n",
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" if not nd:\n",
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" ai, bi = expos[i][0]\n",
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" aj, bj = expos[j][0]\n",
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" pols[i, j] = (z^3 + x^ai * y^aj * z + x^bi * y^bj,\n",
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" (expos[i][2], expos[j][2], 1), ep)\n",
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" else:\n",
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" ai, bi = expos[i][1]\n",
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" aj, bj = expos[j][1]\n",
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" pols[i, j] = (z^3 + x^ai * y^aj * z^2 + x^bi * y^bj, \n",
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" (expos[i][2], expos[j][2], 1), ep)\n",
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" return n, Nf, Ng, Nd, euler, nus, expos, pols\n",
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"\n",
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"def estrato(pol, pesos, euler):\n",
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" Z1 = ZetaFunctions(pol)\n",
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" zetatop = euler * Z1.topological_zeta(local=True, weights=pesos)\n",
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" zetamon = Z1.monodromy_zeta()^euler\n",
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" return zetatop, zetamon\n",
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"\n",
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"def zetas(n, pols):\n",
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" zetatop=0\n",
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" zetamon=1\n",
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" for k in pols:\n",
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" zt, zm = estrato(*pols[k])\n",
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" zetatop += zt\n",
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" zetamon *= zm\n",
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" return zetatop, zetamon / (1 - t^3)^n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"id": "779f6c34-62bc-4484-8caa-f5ef789693a2",
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"metadata": {},
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"outputs": [],
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"source": [
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"n, Nf, Ng, Nd, euler, nus, expos, pols = trinomiales(A, Vf, Vg, Vd)\n",
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"zt0, zm0 = zetas(n, pols)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"id": "5d85cf37-2286-4a0a-a13a-3e5cd781d450",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(-7/39) * (s + 19/39)^-1 * (s + 1/2)^-1 * (s + 1)^-1 * (s^2 - 1207/546*s - 19/14)"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"zt0.factor()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"id": "ae05ef5a-5687-43a8-bac6-7cc9361149d3",
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"2 4\n",
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"1 5\n",
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"6 4\n",
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"3 16\n",
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"26 4\n",
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"13 4\n",
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"78 4\n",
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"39 15\n"
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]
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}
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],
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"source": [
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"for p, j in zm0.factor():\n",
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" print(p.is_cyclotomic(certificate=True), j)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"id": "016e3e91-9506-4223-9010-1b72c474175d",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(0, [76/65/(s + 19/39), -2/39/(s + 1/2), -253/195/(s + 1)])"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"zt0.partial_fraction_decomposition()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"id": "cb10bd5c-a177-4d82-8dec-ce05bd9491bf",
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"metadata": {},
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"outputs": [],
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"source": [
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"n1, Nf1, Ng1, Nd1, euler1, nus1, expos1, pols1 = trinomiales(A, Vf, Vg, Vd, total=False)\n",
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"zt1, zm1 = zetas(n1, pols1)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"id": "d03c12a4-58ef-414a-ba00-e5566f8b2201",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(0, 1)"
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]
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},
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"zt0 - zt1, zm0 / zm1"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"id": "9ee974ba-af17-4998-85a9-d806a92a7242",
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"metadata": {},
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"outputs": [],
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"source": [
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"B = A.delete_columns([3, 4, 11, 12]).delete_rows([3, 4, 11, 12])\n",
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"B[2, 2] = -2\n",
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"B[8, 8] = -2\n",
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"Wf = vector(Vf.list()[:2] + [2] + Vf.list()[5:10] + [2] + Vf.list()[13:])\n",
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"Wg = vector(Vg.list()[:3] + Vg.list()[5:11] + Vg.list()[13:])\n",
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"Wd = vector(Vd.list()[:3] + Vd.list()[5:11] + Vd.list()[13:])"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"id": "e0e38fde-16ba-4e5b-826e-088da1bd1200",
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"metadata": {},
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"outputs": [],
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"source": [
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"n2, Nf2, Ng2, Nd2, euler2, nus2, expos2, pols2 = trinomiales(B, Wf, Wg, Wd, total=False)\n",
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"zt2, zm2 = zetas(n2, pols2)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"id": "de6c6fe8-c3b6-4e9e-b65f-10a41dc2ed05",
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(0, 1)"
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]
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},
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"execution_count": 13,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"zt0 - zt2, zm1 / zm2"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"id": "f09d5d7e-c788-4e6d-96b4-8cf999675ff8",
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SageMath 10.3.rc3",
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"language": "sage",
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"name": "sagemath-310"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.11.1"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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