320 lines
6.6 KiB
Plaintext
320 lines
6.6 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": "60263160-66d3-45f7-a6cd-0042e5c5d37e",
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"metadata": {},
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"outputs": [],
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"source": [
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"R3.<x, y, z> = PolynomialRing(QQ, order='negdeglex')\n",
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"R2.<u, v> = PolynomialRing(QQ, order='negdeglex')\n",
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"R1.<s> = 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": "3e206069-21d2-4ac7-80af-0f997622ce0e",
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"metadata": {},
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"outputs": [],
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"source": [
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"f = u^5 + v^6\n",
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"F = z^10 + f(u=x, v=y)\n",
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"Z = ZetaFunctions(F)"
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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": 37,
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"id": "38a0780c-be08-4a0a-a310-55b279c5528f",
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"metadata": {},
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"outputs": [],
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"source": [
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"zt = Z.topological_zeta(local=True)"
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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": 14,
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"id": "e1e6e6cb-1720-4de5-8a9f-751891aa189e",
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"metadata": {},
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"outputs": [],
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"source": [
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"g0 = v^10 - u^30\n",
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"Z0 = ZetaFunctions(g0)\n",
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"zt0 = Z0.topological_zeta(local=True, weights=[11, 1])"
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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": 15,
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"id": "9d2955da-b2cb-4787-b71b-1a435ca6ae5d",
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"metadata": {},
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"outputs": [],
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"source": [
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"g1 = v^10 - u^6\n",
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"Z1 = ZetaFunctions(g1)\n",
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"zt1 = Z1.topological_zeta(local=True, weights=[11 / 5, 1])"
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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": 16,
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"id": "9dce165f-4968-45b0-82bd-9da8322439a9",
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"metadata": {},
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"outputs": [],
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"source": [
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"g2 = v^10 - u^5\n",
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"Z2 = ZetaFunctions(g2)\n",
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"zt2 = Z2.topological_zeta(local=True, weights=[11 / 6, 1])"
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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": 17,
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"id": "69595c54-4c36-47d8-8b69-f3cc40b345b6",
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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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"(1/110) * (s + 7/15)^-1 * (s + 1)^-1 * (s + 14/3)"
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]
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},
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"execution_count": 17,
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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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"g3 = z^10 - x^30 * y\n",
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"Z3 = ZetaFunctions(g3)\n",
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"zt3 = Z3.topological_zeta(local=True, weights=[11, 1, 1])"
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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": 19,
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"id": "c7e0fa8e-f750-4945-aa62-6d7a82fed340",
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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"
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]
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},
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"execution_count": 19,
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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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"zt - (-zt0 + zt1 + zt2 + zt3)"
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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": 20,
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"id": "042d2c61-0ddf-4b13-b1ab-ade1de1c430e",
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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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"(-1/6) * (s + 7/15)^-1"
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]
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},
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"execution_count": 20,
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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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"zt10 = Z.topological_zeta(local=True, d=10)\n",
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"zt10.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": 22,
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"id": "1bcb4435-1f60-49a0-b26a-14bd6750bb9b",
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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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"1/5/(s + 7/15)"
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]
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},
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"execution_count": 22,
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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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"zt5 = Z.topological_zeta(local=True, d=5) - zt10\n",
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"zt5"
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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": 23,
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"id": "52bb0b78-f7a7-4717-9c11-25a4cec9de02",
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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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"1/6/(s + 7/15)"
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]
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},
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"execution_count": 23,
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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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"zt2 = Z.topological_zeta(local=True, d=2) - zt10\n",
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"zt2"
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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": 25,
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"id": "e613a530-bfa1-4fa1-9c7c-c52e12ccc1cc",
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"metadata": {},
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"outputs": [],
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"source": [
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"Zf = ZetaFunctions(f)"
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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": 28,
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"id": "67800a77-a153-4e9d-9320-75d6db28cbbd",
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"metadata": {},
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"outputs": [],
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"source": [
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"ztf = Zf.topological_zeta(local=True)\n",
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"ztf10 = Zf.topological_zeta(local=True, d=10)\n",
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"ztf5 = Zf.topological_zeta(local=True, d=5) - ztf10\n",
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"ztf2 = Zf.topological_zeta(local=True, d=2) - ztf10"
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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": 34,
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"id": "c9ba3ae1-9f6f-440d-961e-a16b3815287e",
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"metadata": {},
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"outputs": [],
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"source": [
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"p = s + 1 / 10"
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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": 47,
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"id": "faed1d1e-b23e-4da7-9fed-dd470f6fd1d9",
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"metadata": {},
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"outputs": [],
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"source": [
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"h = {}\n",
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"h[1, 0] = 1 / 10 / p\n",
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"h[1, 1] = 9 / 10 * s / (s + 1) * (p + 1) / p\n",
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"h[1, 2] = 3 / 10 * s / (s + 1)\n",
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"h[1, 5] = 24 / 10 * s / (s + 1)\n",
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"h[1, 10] = 99 / 10 * s / (s + 1) \n",
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"aux = h[1, 0] + h[1, 1] * ztf(s=p) - h[1, 2] * ztf2(s=p) - h[1, 5] * ztf5(s=p) - h[1, 10] * ztf10(s=p)"
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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": 48,
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"id": "2c8cd3e8-bc4a-417d-99c6-6903442387ba",
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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"
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]
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},
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"execution_count": 48,
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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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"aux - zt"
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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": 63,
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"id": "39ffdc08-85a3-48db-88f6-edb2b6dcd5c8",
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"metadata": {},
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"outputs": [],
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"source": [
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"h[2, 0] = 1 / 10 / p\n",
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"h[2, 1] = 1 / 10 * (p + 1) / p\n",
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"h[2, 2] = 7 / 10\n",
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"h[2, 5] = -14 / 10\n",
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"h[2, 10] = -89 / 10\n",
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"aux = h[2, 0] + h[2, 1] * ztf(s=p) + h[2, 2] * ztf2(s=p) + h[2, 5] * ztf5(s=p) + h[2, 10] * ztf10(s=p)"
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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": 64,
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"id": "52a2a318-3955-4fad-b5ea-83244a0282d0",
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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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"(1/10) * (s + 1/10)^-1 * (s + 7/15)^-1 * (s + 5/6)"
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]
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},
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"execution_count": 64,
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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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"(aux - zt2).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": null,
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"id": "3a775a53-ee34-4241-acca-7e768f3cf51d",
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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 10.4.beta3",
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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": "3.12.2"
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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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