AbstractAlgebra.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# AbstractAlgebra.jl examples"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We begin by starting up the AbstractAlgebra.jl module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[1m\u001b[36mINFO: \u001b[39m\u001b[22m\u001b[36mPrecompiling module AbstractAlgebra.\n",
      "\u001b[39m"
     ]
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "Welcome to AbstractAlgebra version 0.0.8\n",
      "\n",
      "AbstractAlgebra comes with absolutely no warranty whatsoever\n",
      "\n"
     ]
    }
   ],
   "source": [
    "using AbstractAlgebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Univariate polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Univariate Polynomial Ring in x over Integers, x)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R, x = PolynomialRing(ZZ, \"x\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Create a random polynomial of degree 5 with integer coefficients in [-10, 10]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5*x^5-4*x^4+5*x^3-8*x^2+6*x+8"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = rand(R, 5:5, -10:10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10*x^5+5*x^4-8*x^2-x+1"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = rand(R, 5:5, -10:10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-8196951635"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = resultant(p, q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Multivariate polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Residue field of Integers modulo 65537"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = ResidueField(ZZ, 65537) # R = Z/65537Z"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Multivariate Polynomial Ring in x, y over Residue field of Integers modulo 65537, AbstractAlgebra.Generic.MPoly{AbstractAlgebra.Generic.ResF{BigInt}}[x, y])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S, (x, y) = PolynomialRing(R, [\"x\", \"y\"])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "x+y+1"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f = 1 + x + y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "g = f^5;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "h = g*(g + 1);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "66"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "length(h)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Residue field of Univariate Polynomial Ring in x over Rationals modulo x^3+3//1*x+1//1"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R, x = PolynomialRing(QQ, \"x\")\n",
    "K = ResidueField(R, x^3 + 3x + 1) # Number field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[3//10*x^2+1//3*x+3//5 1//2*x^2-1//3*x+6//1 -4//5*x^2+2//7*x+1//1]\n",
       "[1//2*x^2+x-7//6 -1//3*x^2+9//5*x+5//1 -7//6*x^2+3//2*x-5//3]\n",
       "[5//2*x^2+1//2*x-1//6 -7//2*x^2-8//3*x+7//8 -7//10*x^2-8//7*x-4//3]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S = MatrixSpace(K, 3, 3)\n",
    "M = rand(S, 2:2, -10:10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "t^3+(11//15*x^2-104//105*x-64//15)*t^2+(-71963//25200*x^2-91127//5040*x+46537//12600)*t+(8446987//756000*x^2-1399517//27000*x-269947//36000)"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, t = PolynomialRing(K, \"t\")\n",
    "p = minpoly(U, M) # compute minimum polynomial of M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-8446987//756000*x^2+1399517//27000*x+269947//36000"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D = det(M)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Generic tower"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We will compute with a polynomial ring over a residue ring over a polynomial ring over a finite field"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Residue field of Integers modulo 23"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R = ResidueField(ZZ, 23)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Univariate Polynomial Ring in y over Residue field of Integers modulo 23, y)"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "S, y = PolynomialRing(R, \"y\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Residue ring of Univariate Polynomial Ring in y over Residue field of Integers modulo 23 modulo y^3+3*y+1"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "T = ResidueRing(S, y^3 + 3y + 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Univariate Polynomial Ring in z over Residue ring of Univariate Polynomial Ring in y over Residue field of Integers modulo 23 modulo y^3+3*y+1, z)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "U, z = PolynomialRing(T, \"z\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "p = rand(U, 2:2, 5:5, -10:10); q = rand(U, 2:2, 5:5, -10:10); "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4*y^2+10"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "s = resultant(p, q)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Puiseux series"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We verify an identity on Weber functions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Work to an internal precision of 1000 terms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Puiseux series ring in q over Integers, q+O(q^1001))"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R, q = PuiseuxSeriesRing(ZZ, 1000 , \"q\" )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define the Dedekind eta function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "eta_qexp (generic function with 1 method)"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eta_qexp(q) = prod(1 - q^n for n = 1:50)*q^(1//24)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define one of the Weber modular functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "f1 (generic function with 1 method)"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1(q) = divexact(eta_qexp(q^(1//2)), eta_qexp(q))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Define two Puiseux series in terms of the Weber function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "q^1//3-2*q^4//3+q^7//3-2*q^10//3+4*q^13//3-2*q^16//3+q^19//3-4*q^22//3+5*q^25//3-4*q^28//3+8*q^31//3-12*q^34//3+12*q^37//3-14*q^40//3+17*q^43//3-16*q^46//3+17*q^49//3-24*q^52//3+25*q^55//3-28*q^58//3+42*q^61//3+O(q^127//6)"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = divexact(f1(q)^2, f1(q^20)^2)\n",
    "B = divexact(f1(q^2)^2, f1(q^10)^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Verify an identity between A and B to the maximum available precision"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0+O(q^163//6)"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A^8*(B^18 + 2B^15 + 255B^12 - 580B^9 + 255B^6 - 30B^3 + 1 - 256A^8*B^11 + 256A^4*B^7) + 16A^4*B^11 - B^19"
   ]
  }
 ],
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