MA2-ace TD 6

ace-6.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercice 10.1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "A.<x,y> = PolynomialRing(QQ, order='lex')\n",
    "I = A.ideal(y^4*x + 3*x^3 - y^4 - 3*x^2, x^2*y - 2*x^2, 2*y^4*x - x^3 - 2*y^4 + x^2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On retrouve le générateur de $ℚ[y] ∩ I$ à la fin de la base de Gröbner"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[x^2 - 1/16*y^4, x*y^4 - y^4, y^5 - 2*y^4]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I.groebner_basis()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[y^4 - 16*x^2, y*x^2 - 2*x^2, x^3 - x^2]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I.change_ring(A.change_ring(names='y,x')).groebner_basis()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Une méthode alternative de calculer le même résultat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Ideal (y^5 - 2*y^4) of Multivariate Polynomial Ring in x, y over Rational Field"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I.elimination_ideal(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercice 10.2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "A.<x,y,z,t> = PolynomialRing(QQ, order='lex')\n",
    "I = A.ideal(x + y - z, x^2- 2*t^2, y^2 - 5*t^2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[x + y - z, y^2 - 5*t^2, y*z - 1/2*z^2 - 3/2*t^2, y*t^2 + 1/6*z^3 - 17/6*z*t^2, z^4 - 14*z^2*t^2 + 9*t^4]"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G = I.groebner_basis()\n",
    "G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "z^4 - 14*z^2 + 9"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p = G[-1](t=1)\n",
    "p"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "p(z=sqrt(2)+sqrt(5)).expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "y + 1/6*z^3 - 17/6*z"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = G[-2](t=1)\n",
    "q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q(y=sqrt(5), z=sqrt(2)+sqrt(5)).expand()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercice 10.5"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[X + 126734872901588973582/3395909056778368040767*Y^27 - 1059177032400400444180/3395909056778368040767*Y^26 + 2285808466233304718588/3395909056778368040767*Y^25 + 557276448172777618804/3395909056778368040767*Y^24 + 10527015879533829422789/3395909056778368040767*Y^23 + 4534298082785732166884/3395909056778368040767*Y^22 - 21929288596306940961744/3395909056778368040767*Y^21 + 56329160184432659174606/3395909056778368040767*Y^20 - 32621295253356715497026/3395909056778368040767*Y^19 - 25658713786866124611709/3395909056778368040767*Y^18 + 43205590840914603717979/3395909056778368040767*Y^17 - 165768748128993483079350/3395909056778368040767*Y^16 + 28975447087098703454686/3395909056778368040767*Y^15 + 52105040032950372573350/3395909056778368040767*Y^14 - 31519296754205845938329/3395909056778368040767*Y^13 + 162437763572247864806756/3395909056778368040767*Y^12 - 3236556162508197221408/3395909056778368040767*Y^11 - 40884339127343714124472/3395909056778368040767*Y^10 + 11911188972708254939015/3395909056778368040767*Y^9 - 54062861559277191650146/3395909056778368040767*Y^8 - 17328150062224139585228/3395909056778368040767*Y^7 + 10984661610735135556071/3395909056778368040767*Y^6 - 5735303903628078892952/3395909056778368040767*Y^5 + 792223388139721445320/3395909056778368040767*Y^4 + 13714655845428915245369/3395909056778368040767*Y^3 + 785919437095244302380/3395909056778368040767*Y^2 + 1688372769006791192973/3395909056778368040767*Y - 270367022566925300594/3395909056778368040767, Y^28 - 8*Y^27 + 16*Y^26 + 249/2*Y^24 + 24*Y^23 - 64*Y^22 + 2*Y^21 - 757/2*Y^20 - 56*Y^19 + 96*Y^18 - 6*Y^17 + 379*Y^16 + 72*Y^15 - 63*Y^14 + 14*Y^13 - 251/2*Y^12 - 32*Y^11 + 15*Y^10 - 18*Y^9 - 1/2*Y^8 + 8*Y^5 + 1/2]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.<X,Y> = PolynomialRing(QQ, order='lex')\n",
    "I = A.ideal(X^4 + Y^4 - 1, X^5*Y^2 - 4*X^3*Y^3 + X^2*Y^5 - 1)\n",
    "G = I.groebner_basis()\n",
    "G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Y^28 - 8*Y^27 + 16*Y^26 + 249/2*Y^24 + 24*Y^23 - 64*Y^22 + 2*Y^21 - 757/2*Y^20 - 56*Y^19 + 96*Y^18 - 6*Y^17 + 379*Y^16 + 72*Y^15 - 63*Y^14 + 14*Y^13 - 251/2*Y^12 - 32*Y^11 + 15*Y^10 - 18*Y^9 - 1/2*Y^8 + 8*Y^5 + 1/2"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G[-1]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Y^28 - 8*Y^27 + 16*Y^26 + 249/2*Y^24 + 24*Y^23 - 64*Y^22 + 2*Y^21 - 757/2*Y^20 - 56*Y^19 + 96*Y^18 - 6*Y^17 + 379*Y^16 + 72*Y^15 - 63*Y^14 + 14*Y^13 - 251/2*Y^12 - 32*Y^11 + 15*Y^10 - 18*Y^9 - 1/2*Y^8 + 8*Y^5 + 1/2"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q = G[-1].univariate_polynomial()\n",
    "q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Y^28 - 8*Y^27 + 16*Y^26 + 249/2*Y^24 + 24*Y^23 - 64*Y^22 + 2*Y^21 - 757/2*Y^20 - 56*Y^19 + 96*Y^18 - 6*Y^17 + 379*Y^16 + 72*Y^15 - 63*Y^14 + 14*Y^13 - 251/2*Y^12 - 32*Y^11 + 15*Y^10 - 18*Y^9 - 1/2*Y^8 + 8*Y^5 + 1/2"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q.factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(Y - 0.966506296874216) * (Y - 0.721113386166218) * (Y + 0.597428986963397) * (Y + 0.924209668349044) * (Y^2 - 9.64588608545517*Y + 27.2478986510189) * (Y^2 - 2.07381031598562*Y + 1.07569410961252) * (Y^2 - 0.893217025895166*Y + 0.295336342296285) * (Y^2 - 0.779939824160087*Y + 0.428337852683492) * (Y^2 - 0.121266599180110*Y + 1.03967187384805) * (Y^2 - 0.0603056667782709*Y + 0.938887472264107) * (Y^2 + 0.0272947089885237*Y + 1.03310413303256) * (Y^2 + 0.228048118439437*Y + 0.407906285068887) * (Y^2 + 0.525249644063553*Y + 0.343345271720577) * (Y^2 + 1.29090151259232*Y + 0.509689390328851) * (Y^2 + 1.65011569846010*Y + 4.77398949422654) * (Y^2 + 2.01879686263848*Y + 1.01964113877716)"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q.change_ring(RR).factor()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[(-0.924209668349044, 1),\n",
       " (-0.597428986963397, 1),\n",
       " (0.721113386166218, 1),\n",
       " (0.966506296874216, 1)]"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q.roots(RR)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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e3a779rmt43BSU6XVq6XTT5e++kpq3lz6299cVwVEBiEJiHNFRYSk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\n",
      "text/plain": [
       "Graphics object consisting of 2 graphics primitives"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(implicit_plot(I.0, (X,-5,5), (Y,-5,5))\n",
    " + implicit_plot(I.1, (X, -5, 5), (Y, -5, 5)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'avant-dernier polynôme de la base de Gröbner nous permet de calculer $X$ en fonction de $Y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "X + 126734872901588973582/3395909056778368040767*Y^27 - 1059177032400400444180/3395909056778368040767*Y^26 + 2285808466233304718588/3395909056778368040767*Y^25 + 557276448172777618804/3395909056778368040767*Y^24 + 10527015879533829422789/3395909056778368040767*Y^23 + 4534298082785732166884/3395909056778368040767*Y^22 - 21929288596306940961744/3395909056778368040767*Y^21 + 56329160184432659174606/3395909056778368040767*Y^20 - 32621295253356715497026/3395909056778368040767*Y^19 - 25658713786866124611709/3395909056778368040767*Y^18 + 43205590840914603717979/3395909056778368040767*Y^17 - 165768748128993483079350/3395909056778368040767*Y^16 + 28975447087098703454686/3395909056778368040767*Y^15 + 52105040032950372573350/3395909056778368040767*Y^14 - 31519296754205845938329/3395909056778368040767*Y^13 + 162437763572247864806756/3395909056778368040767*Y^12 - 3236556162508197221408/3395909056778368040767*Y^11 - 40884339127343714124472/3395909056778368040767*Y^10 + 11911188972708254939015/3395909056778368040767*Y^9 - 54062861559277191650146/3395909056778368040767*Y^8 - 17328150062224139585228/3395909056778368040767*Y^7 + 10984661610735135556071/3395909056778368040767*Y^6 - 5735303903628078892952/3395909056778368040767*Y^5 + 792223388139721445320/3395909056778368040767*Y^4 + 13714655845428915245369/3395909056778368040767*Y^3 + 785919437095244302380/3395909056778368040767*Y^2 + 1688372769006791192973/3395909056778368040767*Y - 270367022566925300594/3395909056778368040767"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "G[-2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I.variety()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[{Y: -1.00939843131924 - 0.0274944653961864*I, X: -0.262624822031785 - 0.523806714899075*I},\n",
       " {Y: -1.00939843131924 + 0.0274944653961864*I, X: -0.262624822031785 + 0.523806714899075*I},\n",
       " {Y: -0.924209668349044, X: 0.721113386166218},\n",
       " {Y: -0.825057849230049 - 2.02318289822013*I, X: -0.825057716113690 + 2.02318284841370*I},\n",
       " {Y: -0.825057849230049 + 2.02318289822013*I, X: -0.825057716113690 - 2.02318284841370*I},\n",
       " {Y: -0.645450756296162 - 0.305094594389287*I, X: 0.0606332995900556 + 1.01783863005335*I},\n",
       " {Y: -0.645450756296162 + 0.305094594389287*I, X: 0.0606332995900556 - 1.01783863005335*I},\n",
       " {Y: -0.597428986963397, X: 0.966506296874216},\n",
       " {Y: -0.262624822031776 - 0.523806714899069*I, X: -1.00939843131924 - 0.0274944653961864*I},\n",
       " {Y: -0.262624822031776 + 0.523806714899069*I, X: -1.00939843131924 + 0.0274944653961864*I},\n",
       " {Y: -0.114024059219719 - 0.628414512076182*I, X: 0.0301528333891355 - 0.968492787222865*I},\n",
       " {Y: -0.114024059219719 + 0.628414512076182*I, X: 0.0301528333891355 + 0.968492787222865*I},\n",
       " {Y: -0.0136473544942618 - 1.01632567750100*I, X: 0.446608512947587 + 0.309640401851934*I},\n",
       " {Y: -0.0136473544942618 + 1.01632567750100*I, X: 0.446608512947587 - 0.309640401851934*I},\n",
       " {Y: 0.0301528333891355 - 0.968492787222865*I, X: -0.114024059219717 - 0.628414512076185*I},\n",
       " {Y: 0.0301528333891355 + 0.968492787222865*I, X: -0.114024059219717 + 0.628414512076185*I},\n",
       " {Y: 0.0606332995900551 - 1.01783863005335*I, X: -0.645450756296168 + 0.305094594389296*I},\n",
       " {Y: 0.0606332995900551 + 1.01783863005335*I, X: -0.645450756296168 - 0.305094594389296*I},\n",
       " {Y: 0.389969912080043 - 0.525605670018670*I, X: 1.03690515799281 - 0.0228430063789460*I},\n",
       " {Y: 0.389969912080043 + 0.525605670018670*I, X: 1.03690515799281 + 0.0228430063789460*I},\n",
       " {Y: 0.446608512947583 - 0.309640401851943*I, X: -0.0136473544942619 + 1.01632567750100*I},\n",
       " {Y: 0.446608512947583 + 0.309640401851943*I, X: -0.0136473544942619 - 1.01632567750100*I},\n",
       " {Y: 0.721113386166218, X: -0.924209668349044},\n",
       " {Y: 0.966506296874216, X: -0.597428986963396},\n",
       " {Y: 1.03690515799281 - 0.0228430063789459*I, X: 0.389969912080024 - 0.525605670018678*I},\n",
       " {Y: 1.03690515799281 + 0.0228430063789459*I, X: 0.389969912080024 + 0.525605670018678*I},\n",
       " {Y: 4.82294304272759 - 1.99677716774417*I, X: -56.0180043338967 + 43.8237865954014*I},\n",
       " {Y: 4.82294304272759 + 1.99677716774417*I, X: -56.0180043338967 - 43.8237865954014*I}]"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "I.variety(CC)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Exercice 10.6"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[u - z, v^2 - y*z, v*x - z^2, v*z^2 - x*y*z, x^2*y*z - z^4]"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.<u,v,x,y,z> = PolynomialRing(QQ, order='lex')\n",
    "I = A.ideal(v*x - u^2, u*y - v^2, z - u)\n",
    "G = I.groebner_basis()\n",
    "G"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "L'idéal obtenu par élimination contient une hyper-surface parasite $z=0$. À l'aide d'une variable $t$, on peut forcer les dénominateurs $u$ et $v$ à ne pas s'annuler en ajoutant le polynôme $1 - uvt$ à l'idéal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[t*x*y - 1, t*y*z^2 - v, t*z^3 - x, u - z, v^2 - y*z, v*x - z^2, v*z - x*y, x^2*y - z^3]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.<t,u,v,x,y,z> = PolynomialRing(QQ, order='lex')\n",
    "I = A.ideal(v*x - u^2, u*y - v^2, z - u, 1 - u*v*t)\n",
    "G = I.groebner_basis()\n",
    "G"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[t + 1/2*x^2*y - 1/2*x^2 + x + 1/2*y + 1/2, u + 1/2*y - 1/2, x^2*y^2 - 2*x^2*y + x^2 + 4*x*y - 4*x + y^2 + 3]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.<t,u,x,y> = PolynomialRing(QQ, order='lex')\n",
    "I = A.ideal(t^2 + t + 1 - x,\n",
    "            (t^2 - 1) - y*(t^2 + 1),\n",
    "            1 - u*(t^2+1))\n",
    "G = I.groebner_basis()\n",
    "G"
   ]
  }
 ],
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