Polynomials.Solving.Resultant.Theory.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "5cb60ede",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( x, \\  y\\right)$"
      ],
      "text/plain": [
       "(x, y)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "from IPython.display import display\n",
    "init_printing(wrap_line=False)\n",
    "var('x,y')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "82fa1c36",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 3 x + y + 5$"
      ],
      "text/plain": [
       "-3⋅x + y + 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + y^{2} - 5$"
      ],
      "text/plain": [
       " 2    2    \n",
       "x  + y  - 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - x^{3} + 3 x^{2} - 3 x + y + 1$"
      ],
      "text/plain": [
       "   3      2              \n",
       "- x  + 3⋅x  - 3⋅x + y + 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = y-3*x+5 \n",
    "g = x**2+y**2-5\n",
    "h = y-x**3+3*x**2-3*x+1\n",
    "display(*[f,g,h])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "1c346080",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "'\\nplt1=plot_implicit(f, show=False, aspect_ratio=(1, 1))\\nplt2=plot_implicit(g, show=False, aspect_ratio=(1, 1))\\nplt3=plot_implicit(h, show=False, aspect_ratio=(1, 1))\\nplt3.extend(plt1)\\nplt3.extend(plt2)\\nplt3.show()\\n'"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot_implicit(Or(Eq(f,0),Eq(g,0),Eq(h,0)), aspect_ratio=(1, 1), \\\n",
    "              annotations=[{'xy': (-2, 2), 'text': '$g$'}, {'xy': (2, 4), 'text': '$h$'}, {'xy': (3.5, 4), 'text': '$f$'}] )\n",
    "#Much better than \n",
    "'''\n",
    "plt1=plot_implicit(f, show=False, aspect_ratio=(1, 1))\n",
    "plt2=plot_implicit(g, show=False, aspect_ratio=(1, 1))\n",
    "plt3=plot_implicit(h, show=False, aspect_ratio=(1, 1))\n",
    "plt3.extend(plt1)\n",
    "plt3.extend(plt2)\n",
    "plt3.show()\n",
    "'''"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "297ef81f",
   "metadata": {},
   "source": [
    "<font color='red'> <b> GCD isn't showing that root exists b/w polynomials </b></font> "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "935cefbb",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1 1 1\n"
     ]
    }
   ],
   "source": [
    "print(gcd(f,g), gcd(f,h), gcd(g,h))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4cce4c3c",
   "metadata": {},
   "source": [
    "Roots of tuple pair show there is interception among pairs of polynomials; it doesn't show there is a <b>common interception</b> among x3 polynomials <br> The last x3 element tuple does show common interception among polynomials"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c7a6c211",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(- 3 x + y + 5\\right)^{\\frac{1}{x^{2} + y^{2} - 5}}$"
      ],
      "text/plain": [
       "                   1     \n",
       "              ───────────\n",
       "               2    2    \n",
       "              x  + y  - 5\n",
       "(-3⋅x + y + 5)           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(- 3 x + y + 5\\right)^{\\frac{1}{- x^{3} + 3 x^{2} - 3 x + y + 1}}$"
      ],
      "text/plain": [
       "                          1            \n",
       "              ─────────────────────────\n",
       "                 3      2              \n",
       "              - x  + 3⋅x  - 3⋅x + y + 1\n",
       "(-3⋅x + y + 5)                         "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x^{2} + y^{2} - 5\\right)^{\\frac{1}{- x^{3} + 3 x^{2} - 3 x + y + 1}}$"
      ],
      "text/plain": [
       "                         1            \n",
       "             ─────────────────────────\n",
       "                3      2              \n",
       "             - x  + 3⋅x  - 3⋅x + y + 1\n",
       "⎛ 2    2    ⎞                         \n",
       "⎝x  + y  - 5⎠                         "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(-1\\right)^{\\frac{- 2 x^{3} + 6 x^{2} - 6 x + 2 y + 2}{x^{2} + y^{2} - 5}} \\left(- 3 x + y + 5\\right)^{\\frac{1}{x^{2} + y^{2} - 5}}$"
      ],
      "text/plain": [
       "         3      2                                          \n",
       "    - 2⋅x  + 6⋅x  - 6⋅x + 2⋅y + 2                    1     \n",
       "    ─────────────────────────────               ───────────\n",
       "              2    2                             2    2    \n",
       "             x  + y  - 5                        x  + y  - 5\n",
       "(-1)                             ⋅(-3⋅x + y + 5)           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(root(f,g), root(f,h), root(g,h), root(f,g,h))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e26775cd",
   "metadata": {},
   "source": [
    "Representation varies"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "27433291",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x^{2} + y^{2} - 5\\right)^{\\frac{1}{- 3 x + y + 5}}$"
      ],
      "text/plain": [
       "                  1      \n",
       "             ────────────\n",
       "             -3⋅x + y + 5\n",
       "⎛ 2    2    ⎞            \n",
       "⎝x  + y  - 5⎠            "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(- x^{3} + 3 x^{2} - 3 x + y + 1\\right)^{\\frac{1}{- 3 x + y + 5}}$"
      ],
      "text/plain": [
       "                                1      \n",
       "                           ────────────\n",
       "                           -3⋅x + y + 5\n",
       "⎛   3      2              ⎞            \n",
       "⎝- x  + 3⋅x  - 3⋅x + y + 1⎠            "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(- x^{3} + 3 x^{2} - 3 x + y + 1\\right)^{\\frac{1}{x^{2} + y^{2} - 5}}$"
      ],
      "text/plain": [
       "                                1     \n",
       "                           ───────────\n",
       "                            2    2    \n",
       "                           x  + y  - 5\n",
       "⎛   3      2              ⎞           \n",
       "⎝- x  + 3⋅x  - 3⋅x + y + 1⎠           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(-1\\right)^{\\frac{- 6 x + 2 y + 10}{x^{2} + y^{2} - 5}} \\left(- x^{3} + 3 x^{2} - 3 x + y + 1\\right)^{\\frac{1}{x^{2} + y^{2} - 5}}$"
      ],
      "text/plain": [
       "                                                    1     \n",
       "    -6⋅x + 2⋅y + 10                            ───────────\n",
       "    ───────────────                             2    2    \n",
       "       2    2                                  x  + y  - 5\n",
       "      x  + y  - 5   ⎛   3      2              ⎞           \n",
       "(-1)               ⋅⎝- x  + 3⋅x  - 3⋅x + y + 1⎠           "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(root(g,f), root(h,f), root(h,g), root(h,g,f))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f0e6c089",
   "metadata": {},
   "source": [
    "# Sylvester’s Resultant $(n=2 \\ m=3)$\n",
    "By calculating the <b> determinant of Sylvester’s resultant </b> we know that the system has a common root. That is because the determinant is equal to 0 (or can be fully factored). <br> When we have a system of <b> m polynomial equations in two n variables </b> we can solve for one variable where the second one is kept as a coefficient. This actually allow us to find the roots of the system. That is also why the resultant is often refereed to as the eliminator <br> https://nikoleta-v3.github.io/posts/resultant-theory/ <br> Further read about Macaulay Resultant https://github.com/Nikoleta-v3/resultant-theory"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "1f5a0223",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}1 & 5 - 3 x & 0\\\\0 & 1 & 5 - 3 x\\\\1 & 0 & x^{2} - 5\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1  5 - 3⋅x     0   ⎤\n",
       "⎢                   ⎥\n",
       "⎢0     1     5 - 3⋅x⎥\n",
       "⎢                   ⎥\n",
       "⎢             2     ⎥\n",
       "⎣1     0     x  - 5 ⎦"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.polys import subresultants_qq_zz\n",
    "matrix = subresultants_qq_zz.sylvester(f,g, y) #matrix in terms of x, only accepts x2 polynomials\n",
    "matrix"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "893cb85b",
   "metadata": {},
   "source": [
    "factor reveals common root at x=2 & x=1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "9f098213",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 10 x^{2} - 30 x + 20$"
      ],
      "text/plain": [
       "    2            \n",
       "10⋅x  - 30⋅x + 20"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 10 \\left(x - 2\\right) \\left(x - 1\\right)$"
      ],
      "text/plain": [
       "10⋅(x - 2)⋅(x - 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ 1, \\  2\\right]$"
      ],
      "text/plain": [
       "[1, 2]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(*[matrix.det(), factor(matrix.det()), solve(factor(matrix.det())) ])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6619b4f6",
   "metadata": {},
   "source": [
    "Solve for corresponding y for each known x for each polynomial. We have <b>x1 common root (intercepts) </b> $(2,1)$ in all polynomials  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "bb7a27aa",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - 3 x + y + 5$"
      ],
      "text/plain": [
       "-3⋅x + y + 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 1, \\  \\left[ -2\\right]\\right)$"
      ],
      "text/plain": [
       "(1, [-2])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 2, \\  \\left[ 1\\right]\\right)$"
      ],
      "text/plain": [
       "(2, [1])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + y^{2} - 5$"
      ],
      "text/plain": [
       " 2    2    \n",
       "x  + y  - 5"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHgAAAAUCAYAAABGUvnzAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjYuMCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy89olMNAAAACXBIWXMAABJ0AAASdAHeZh94AAADw0lEQVR4nO2ZXYhVVRTHf5M+JFIJIyQomiHaQ5YRQT5kilqIFKb5VKMihAUxGAkhGP/5F1kPlfUQRQhOD4EFA4ka+VLaB/QlCYVlFE0ZGZMYmZnYx/Sw97mee+bMeOeeM/fD/MOw7137zvqvddbZe6+1dsfg4CAXceHikmYbcBFji/F5QtuvAMuAmZL+aKxJVXYsBN5JiY5IuqY51jQXticDv6Rlkjri3I3Ap8B9kranfzMkwLZvArqATeng2r4buBWYB1wPXAa8KuneUj3JxwFgP3C8TKW2O4G7gOXAXGAqcBb4HNgB7JD0b5mcBXhPA46f1wEzkglJB22/ATxue6ekU8lc3gp+AjgJvJiRbyEE9hTwI9DIlbRfUs8Y6F1N8PMYYaf4AbgSWAlsB5bZXi2p7ERl1LySTgM9UNnZZlCNJ4GPgG5gayKsOoNtzwaWAK9L+jOj4CFgNnA58EAB51oJXwN3AtMk3SNps6T1hJf3KLCK8NBbnlfSx8BXwAbblbhmV/B6oAN4LUdB5Sy0nZ1uS0h6exj5z7ZfIuxmC4G+NuHdSVjlS4F9MDSLXgL8A3w4SsUXIv6K499txPtBHJcmgkqAbU8kJFBfNjNzbgXYHg+siV/faiPeT+K4IBGkV/BUYBzh4P+/4yngWuBNSfvahVfSb8AZYHoiS5/BnXH8tYiFjYbtfoZmlCNhxNLOdjfwMCFh6SpmXe0okfcEISMHqgOcZM2XFlDeDHxLeGtrxU/DTdh+EHgeOAwslnSioG01oWTeCZyLZVWAB+LYSRtB0uIy9NjeCGwDviA85IGR/6MclMkby6NJwHeJLH0GHyO0wubUS9CusP0I4SEfAhY1MLhl884hlLmHEkElwLFr8i4w2fasgkQA2O61PWh7XRn6xgK2HyUkNwcJK+i87dAy/KqHtwbcHMdKzyLb6OgjdFFuB77JGLQCWBG/TonjfNu98fNxSZsy+pIXqNG1ZE2wvRZ4jFD7vwd05zRx+iX1ZmSF/CrAez7cFnXuSgR5AR4g1GIvZObmAWszsqvjH8D3QDbAc4Hfgb2jNLRRmBnHccDGYX5zAOjNyIr6VS/vsLB9BWEB7pF0NJFXBVjSWdvPAVtt3yDps9RcD7HZXSPhJOA64BlJLVl6jdYnKMevenhrwBpCBfR0Wph3m7QNuJ+whdxRgPAWQtvt2QI6Esi2aI374DL9qhl598GpuQnAZqBP0vvpuSEBlnTGdhewyPbEetuWknZTvKbu59wdKJR8H1wPSvKrHqTvg7O4CniZnC39P2DOfqXqEGDpAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left( 1, \\  \\left[ -2, \\  2\\right]\\right)$"
      ],
      "text/plain": [
       "(1, [-2, 2])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 2, \\  \\left[ -1, \\  1\\right]\\right)$"
      ],
      "text/plain": [
       "(2, [-1, 1])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle - x^{3} + 3 x^{2} - 3 x + y + 1$"
      ],
      "text/plain": [
       "   3      2              \n",
       "- x  + 3⋅x  - 3⋅x + y + 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 1, \\  \\left[ 0\\right]\\right)$"
      ],
      "text/plain": [
       "(1, [0])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 2, \\  \\left[ 1\\right]\\right)$"
      ],
      "text/plain": [
       "(2, [1])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for eq in [f,g,h]:\n",
    "    display(eq)\n",
    "    for _x in (1,2): #solved x values\n",
    "        y=solve(eq.subs(x,_x))\n",
    "        display((_x, y))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "dbc0398b",
   "metadata": {},
   "source": [
    "single common intercept of f & g & h"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "8939244a",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left\\{ x : 2, \\  y : 1\\right\\}\\right]$"
      ],
      "text/plain": [
       "[{x: 2, y: 1}]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve((f,g,h), dict=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "21459b75",
   "metadata": {},
   "source": [
    "x2 intercepts of f & h (see graph)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "53d01128",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left\\{ x : -1, \\  y : -8\\right\\}, \\  \\left\\{ x : 2, \\  y : 1\\right\\}\\right]$"
      ],
      "text/plain": [
       "[{x: -1, y: -8}, {x: 2, y: 1}]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve((f,h), dict=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "89c9c742",
   "metadata": {},
   "source": [
    "# Dixon Resultant $(n=3 \\ m=3)$\n",
    "By calculating the <b> determinant of Dixon’s resultant </b> we know that the system has a common root. That is because the determinant is equal to 0 (or can be fully factored). <br>When we have a system of <b> m polynomial equations in n variables </b>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "8c02bccd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( x, \\  y, \\  z\\right)$"
      ],
      "text/plain": [
       "(x, y, z)"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy.polys.multivariate_resultants import DixonResultant\n",
    "var('x,y,z')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "66359e87",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + y^{2} - 1$"
      ],
      "text/plain": [
       " 2    2    \n",
       "x  + y  - 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + z^{2} - 1$"
      ],
      "text/plain": [
       " 2    2    \n",
       "x  + z  - 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle y^{2} + z^{2} - 1$"
      ],
      "text/plain": [
       " 2    2    \n",
       "y  + z  - 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "f = x ** 2 + y ** 2 - 1 + z * 0\n",
    "g = x ** 2 + z ** 2 - 1 + y * 0\n",
    "h = y ** 2 + z ** 2 - 1\n",
    "display(*[f,g,h])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "f4292fc0",
   "metadata": {},
   "outputs": [],
   "source": [
    "dixon = DixonResultant([f, g, h], [y, z]) #in terms of x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "af88cf40",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA44AAAAZCAYAAAB+fHDyAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjYuMCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy89olMNAAAACXBIWXMAABJ0AAASdAHeZh94AAAXSUlEQVR4nO2dedgcVbGH37iwXMMmqOQqqwiooDFBFASSCKiIXkFFvCAQXFgiSEAQAkjxCwIuQCK7gBBAeUCJooIrCEiI7ITlioBIEBREFllEiEDuH3U66a+/7pmeme7pni/9Ps/3TNKn53RNd3V116k6dUYtXLiQhoaGhoaGhoaGhoaGhoYsXlG1AA0NDQ0NDQ0NDQ0NDQ315lVVC1AkkkaZWRNCbYOkacDHgfWAF4DrgGlmdmelgjXUhkZHRg6DaBeXFP0bxGtTJ5YUPWloaGgom7zPoxETcZS0BrBN1XIMCBOBU4FNgfcDLwKXS3ptlUI11IqJNDoy8EjaElirajm6YCJLhv5tJ2mVqoUYYCayZOhJQ0NDQ9l8SNLq7XYaNRLmOEpaDjge2MvMXq5ankFD0mjgKWA7M/tZ1fI01I9GRwYPSW8BdjIzVS1Lr4xU/ZO0FHAisI+ZvVi1PIPOSNWThoaGhrKR9ArgdOAAM3s2a7++papKWhO4HzjXzCYX3P1hwNlZTqOkc/Fo5Fpm9q+Cjz0SWA6PPj8ZbZA0HrgJ+IKZnVWVYIPMCNO7YToyqCxBun0s8NmqhSiIEaN/ccxsgaRfAl8ATqtanhHAiNSThsFA0mrAX4Afm9nHq5anGyQdgAdidjazC6qWp2gkTQSujG2628zWr0aa/hKyW/4R32Zmo2L/flnS2cCh4S+VIY6jpGT48WXcAN8OnFVHJZI0BtjSzA7JaH83sAtwYPLlXdIngQnAWOCd+EPn+2b2mVKFzkDSysD2wLbAhsAbgQXAHcA5wDklRVS/DcwDfh9tMLObJV0CHCXpwlajDw3DaaV3BfRdhZ4M05FBZUnQbUnvA/5tZk8X3G9tbNQI4jLgRkmzzOzfVQtTFI2dalgCGR8+b6lUit7YKHzeVKkU5XM1cBXwWMVy9JPngCgDaTKwRnIHM7tO0kmSTjKzh9M6yYo4Rh2/Glgf+BgwSdJGZnZAT2IXz4HAj1q0Hw08Tfpo7uG4w/gs8BD+W6tkB1zOh/ERkb8Ab8An/58FbCNphyKLKUg6AdgM2MzMXko0HwtcD3wJOKaoYy4htNK7XumrnrTRkUFlpOv2ofjcr6Kpm40aeMzsP5Lm4tHhU6qWp0AaO9WwpDEufN5cqRS9cQgwHbi3akFaIWl54FfAqIxdlsHf6ecBSwNvN7OlYu1XmdmRZcpYN8zsOeBIWBR5HeY4Bn4MfAXYP60x1XFMnsxQYOE3wFRJJ5rZ/M5FLp6Qj7tT+EtrXxfYCo+Wpo3k7o87jH/CI49XpuzTT+4B/ge4LD4SK+lQ4AbgE/hDd3YRB5M0A/g0MMnM/pxsN7MbJP0R2FPS15eU+aOSJuMj4pPM7Kouvt9O73qlb3rSTkeKptdzn5c663YB+rcy8AFgt2IlA2pmo8qgXzqY4HpgD0aW4zhi7VSnVKRTDf1n4COOZvaXqmXIQ8im2SStTdKrgYsBwzM6LsIHixvycR1wgaQvp70b5ZrjaGZXhJestwLvBuZHbZI+BeyDR+6Wwp2wC4ATzOyFrD4lrQ/chXv9kzL2uQMfMVg9I2S6IbAqcGvGYT6Lj0ZclPG7FjmKUvX1I8zstxnbH5F0Oh7FmkjiQSvp18DWwCfNbHZs+yj8YbUb8I14Oq+kbwM74g+yP7YQ60J8hGJrfHRnCOEG/SIe9l4PeBz4IXAwrl/3A5eb2c4tjpFKmX2XTEu965Vu9KQsHZE0B3hfC3GvMbMtcv60nuhClsp0u2QmAI+aWeEpOHWyUbG5OAea2fEp7evh0yyu76MOdivTPOA9kkYnU6cHVQ/rYqe6uSZ1sWt1kr0OelgTGV4VZPgcsC7wdzy745t4xPFvZvb32P6jcL3dHdgAeA3+7jvTzM5N6X8S8Fv8ul8AHAFsAbwybN873ENvw7PmtsIjatcAU5KOn6Sd8HTxdwNj8MrDfwJONbNzMo59nJkdFNu+NfBr4OvA9/GMli3Db7kd2N/Mrs99EkskBJTOx/Xip7jd+IWZ/aCDPjq2Qzn6LO15VcI9fyueHbIhcFuysZPlOKJw8KK0EknH4C/Hb8UV/OSw3zHAr0LFuFSCgb8SmBgiNEOQtCl+k/0kK88Wz8V+1Mz+mdG+FfAS7j0POv8Jn2mV9w7C56MeJemVse3H4Qp+RuJBewpuxHYCnpS0avgbndL3teFz62SDvOT5tcAM4EG8OuBTwFQ8BXAv4LX4qE9HlNl3H6hS77L0pCwduQRPbU/+RQ+v1JfHkuhUlkp0uw+Mx6M9/abfNiq6fu/NkOck/GVrn45/Sfd0K9PdYfvY+MYB18NW9NNOdXNNLqEedq0WstdBD2siw1LAz4GZ+DP+JOAKfADyDDyQcXNs/2WBX+LOxorAucDZYb9ZIfqeJEp3XReYE47zXfz6bQecLemjeNR+dOjzHtw5PC8h73Jh29q4Y3kyPrVrrdDPwRnHTkZM3xWT6cbYca/Go34/DceqlODYnQn8zsy+h1+nO6zzIngd2aGclPm8uoQC73kzexJ4lMXzXYeQK+IoaSt8dGchrjRI2gSYht/AG5vZI2H7NDw/9iP4/MNW84dOBSbhKToHJtr2CJ/fafH9NwP/zJD5NfhD+K6ii5P0mzDCtWv47y+T7WZ2m6TzcYXehcUG6QDgB8Deia9MCZ9XJA9FyH+OcWP4TButuAgfxdrPzE4Msn4LT//dBngbMMvM/tTq92VQZt+lUaXetdKTsnTEzI5LkeM4YHX8YTm9m9/SDV3IUpVul81a+AtV36jIRt0C/Bt4T4o8O+ADAiea2e2d/p4e6EomM3te0gL82s2JNQ2yHqZSgZ3q+JrUyK7VRfY66GEdZDgFP+dHAF+zMD9X0izciYKhTtcF+LSBQ81sUaqkJAP+CBwh6bTwoh4ROW8bA++Nrq2k6bgj8AHckdvazH4f2qJsvy0kLWNmz4c+FgJvit7PY8c/HHc2dwe+kXLspOMYbd8ceJ+ZLWqXNBtPOR+LO6fDkDQVd5zzMs/MLulg/4gZwH1mdqqkI4Gn06J77ejCDuWhtOdVSff8U/iAwzBSHcdwwsGL46yHj3KMAmaY2QOhLSrz/rW4UprZi5K+DHwY+DytHcdL8InzkyUdFqW2SloR+BRwH3B5i++vhBcgSeONuPeeFa0cJL6OR19/bmbDUuoCX8XTdSyMth6Np9/tksxRtlj53XaY2VOSnscVcBFhMGEr3FCcFNv/MUnz8cWYX6ALhS2z7z5Qpd6105NSdCQijPadghvVU4B9rcAiKUXLUoVu94mV6H+luL7bKPOiMjfiL0tjosyUMHhzAj5iekS3P6gbepTpGfzaEb4z6HqYRV/tVK96UqVdq4PsddDDmsiwMf5Oe6mZHRVvM7PfSboLz767Oez/Efzd+YdxpzHs/w9Jl+IDKOMYOvAROWmTEwMCz4TfOhY4KHIaQ9sCSXcDq+Hpo8+H7c/ixR+HYGYPS/obHqGNMy7sn8xYiSKOu8edxsBd4XOZ5HFiTCW7EEsa5+L+QW4kHQUsMLNjJO0HrGxm+3bSR4LcdigP/XpeFWivnib2PIqTFXGMQv0L8YjeNcB3Q+g3IlLuYSFQM7tH0kPAWpJWMLPU0e/gZJ6Jn6xP4KMz4B7+sng4uNUPXopwg6SwcvgsdT2ncCN3ckN0tNyHpC8BX8ZHp3bJ2s/MHpQ0E6+IdRIwF/i4mS3oQLYsnsDzneNEssxMuUbRNfmOmT0Yb5A0BU8DGAP8HzDVzJKjVB33LWkLPGo9Hvhv3MDNavfD2ly/KzV87mu7dUjb6l0ZOpNHT8rUkZDOcTb+IPymmSVTYNK+M59iz303svRbt5Oyzqf4c9DKLhaufxXbqGvxiPEmLK6ufQTwJtwGtIy8lnT+u5XpefzaRXSrh2XYwjQ6XrqqQjvV1TWpiV3rm+wZ9M0eFilDt/dBCyIn5OiM9sfDZ+RYRVlz30jZN77/olTI4ESsC/zZzIZlbuB69QTp9RPWAJ4xs6hfJK2Epz5uiweBlmfoFLVbY/tGx54bP8fBaXoL8ACeppskikrdl/orATNbM6utCCQdCLzezPaUtBvun0zupc862aG8FHjPw/Dn0SKyqqrmiTasED6zIisP4yP5K9I6beoM4DBgTxY7jnvg6z2dk/WlwHP42otpRNUsW42CFMF9tHhJS+FveXeUtA++LtUf8LUqn2jzlfjCnp8zL71bBMuy+HxGTMDnqaQZN/BrMyTaLGlH/PdMwdOxpgC/kPQ2Gzqhu+O+8Zz7O/F8/vOGfSObmQxPoRiLL0FzLrFCUIF5bfrLo3eF6kyHelK4jsgLFlwAfBIwM8s76juTYs99N7L0W7eTzKTgcxDkS5uvHFGY/tXARkXzRt4D/EhedG1/fB2/YYUnUphJ8ee/W5mWw69dRDd2ELq3haU9y6ByO9XxNamRXeun7Gn00x4WJgPd3wdZfAB39rKKwKyN19z4a/j/BOARM8tammNM+Iyfj3fijt2wTDtJa+IRoB+Z2X8SbaOBdYiluUt6B17Q5g34fMgLg/wv4inxuzK08El07GREcSyecfibjEDOOPwd//6M31kqkvYMMnxG0vbAR4FPF5QVULkdykvB9zwMfx4tItccxwwiZ3BV0kcaxiT2S8XM/irpp8D24SS+Fk9jucjM/tHqu3g1q+Uz2h4NnytntBeCmW1ZRr/ynPAZuOHb0swebbP/TvjE3Ufwa7If3eVhJ/t9Bf4QvD+2bVl8UOC+5I0kaW28Eu5ci1UWCxyAz0E4M/x/X0kfCnJO66VvM/s5YTRMPt8gF2Y2M+U3T8Yf8rOs89LpbfWuSJ3pRE/K0BFJy+Blr7clo1pYFkWf+05l6bdup1GC/oHr4KotjlmI/tXERs3FM2OiggMn4yP4X8zz4lDS+e9YppBeNBo/N73oYS+2sJRnWZBjKhXaKTq8JnWya/2UPaWvvtrDImXo9j7IkGEZ4PXArRnnfBwe1fxF+P9y+Ltp6nzLEBmagNvqu2NNUSZfmrM5vkXbu3DnLu70nY8/34YtASOfLwlwU8qxb2UomTKF37kucHUre6uS5jhK2hnX80/g9VI+j0cF0wq0dUQd7FBeirznYyxPeB4l6aSqapJIuSYmGyStg4de77fsiqdxooWq9yRfUZyIe4BVMtoexkcL1svRT62QV7qagY9ETsrxQvZhYBb+UH4Hbog+Ly/v2yvr4QZpXmzbsmFbWp73DLw09JAbVz55ezw+Ahbn18CmvfRdM/qmd53oSRk6ElJbLsPnM08pyFj1U5Z+63a/uJeSB8zqYqPMi0rcBYwPD/ot8ZS15MtP3+hSppXw53E0t2jQ7eAiqrZT0Nk1qZNdg8plr4M9rMO98FL4e31G+2HhM3LcXsAdhNdl7D8ZD66ck3AYIiftpmHfWOw4prVFcxBvAZC0Gn7/XJXiNK6IF8WBoc5gu4qqWccdRbozG2cqPgUu7992bfpD0sfw8/ipIPs0YEdrsQxgXupghzqQtSx7tTJDBzUW0YvjeHb4PFzSopsjjKQcF/r+bs6+rsAfmLvhSnC3xdZYbMFcYLkQwh9CuBl/B6wSHNmBQNJX8eIBN+Mjsy2LXEjaDB9peAj4YIjSHo5Hk7Ny6zshGhmJX48n8QnU64R0iEiWvfEFn2H46NIq+MhKcmTy7wyNjnTTd23ol951oidl6IikFfAXgQn4JP7TuumnCHqQpd+63S+uA94eoliFU0MbNQcvCPEdvCjQYa137wudyrQhnp3zh/D/gbaDEVXbqQRtr0md7FqCUmSXNEvSwhARTaN0e1iSDLnJcXxCaui9wBvlS2HEv38wXlUUggMV5sHdAKwmL+wT339LPI13PsMXpR+HT9O6M0WMVhHHpNMXpZyvHVIYo2OvjM+PfBPubM9L9PE8i21QUqa0ap+RTElncwhmtqaZjergb3Kr/sI5PRA/7+vg9mEHS6yB2w3d2KE8OhQj97OhXb9l2StJa+GpqnPT2rtOVTWzuZK+CXwFuFPSxcC/8LLIG+An51s5+1ooXxD4hLDpjJzfe0jSbfiox/yUXWbjIewPkpIyIGk7Fo9sRAZtk1haw2NmllwmpDTkk3qn4yNb1wBf0vCJ9PMtTPCWNBa4FH/h2NpClSYzu1jSTcDHJG1u3U9IB8/rfwn4SbQhXK9Z+KTryyX9ED9/24f9VsDX5zwdL6p047BeMyiz7z7SUu96pRM9KVFHLsBHj2/AH05HpuxzbBGjfyXK0lfd7iNz8ZeCN1Ow/tXURl2LZ6qMxheifrKHvoqiU5nG4otUvwwjQw9rYqfi5LkmdbJrccqSPQoepEbs+qSHVcvQ8vgxjsXnoc2WdCGexjcRH/R5EK9oGnegDsOrcF4q6Qf4nOB34u8FD+B6vmgql6Sl8SVFbrf0AizjgQcsVvwmxjh8rv5dsKhq62/xarPXS7ocj3Bugw+Uvgz8wcKyHbFjz4uneeaUCdpHHAtDvsb70XiE7XXAabjT2LPd78EO5dUh6OzZ0K7fsuzVu4Cbo9+fpJc5jpjZwZJuxW/oXfHlO+7DvfPjMxQti1l4pHIBnU0SPQMv0/zjlLbZeA75rnhZ2iRj8ShnnLVZXCXqAYavL1kma4XPV+Kh/TSuxteUWQefKL4QHxVJzjOdBvwGd96zFhxtSRjN2A4vP/1govkgPB1jR/wmeBJPGzkEV7rz8NTj2WH/x/AXiGQFyzcwPI+6077rRju965VceiJpDiXoiHxu4BbhvxuHvySPmtmRefvslm5lqVC3S8e8NPv5+EtD0QMXtbJRgWiO6o3kz3Ipm05lej+xZQYCg24HK7VTKbS8JnWyaymUJfuG+DIwl7U4dtn2sAwZOiHP8TGz8+RVSvcD/hevbjoH2At//3zcFi9Xh5ldEaKLxuKI5P3AUcBxZvZM4hAb4O/QaXMJ18BTB69OaVuasAyImb0Ua/o0cDzuLO4N3IHPPb0DL6ASTz2Njp2MHG6A+wlpaargjmPa8h1lchaeHvsTPNr4DHBxyqBUxL1mtntWY0SPz6tcOhTo5NmQ2W/J9mor4MysxlELF3Y9H7NQJE3ER0K+Z2aZJd1TvvdqfIR9szSvWtI0vNrWuCrnvQwikvYFTgQ2N7M57fbP0d/1wG1mtkds2z3AbDPraMJ8m+M8C+xjvZXe7lWGRu9qzKDqdgfyjMGLU3yw38fuN/Liatvii2XXIvLWiUySVgEuNLOtWu3XpRyV28K6UEc9yUsZssvnuj2OD/J/pYg+Y33nsodlypCQJ/U+6NfxG+pDzNdQEYNAnepQ3nu5bN2UdBUwwRKraISBiDnAppao3hvRU8SxYKITc3InXzJfVPN4YGcWz7uMMwMfEZqOl+ltyIG8ktk03ND3/GIdOAE4X9INeLh+L7wS2em9dqzF5ajBw/urh7SDJ6y78t+90uhdTRk03e4G8wWeb5G0gZmlzZcZEcgLDHwUOKUuzkAXMn0OyBwu7+L4dbOFlVNHPclLibJvji9xcUK7Hbsgrz0sTYac90GZ56Ch3pgkw2uqrN9DP7l1qMN7uXDdDIOU7Var2BmPiKc6jVCx4yhpQ+AjeLh7GzxtLGuNnEzM7EJJ0yWtYonJ92b2vKRdgEmSXmNm/ypE+JHPmnga8KyiOjSzi+STsw/H8+3vBD4cT+/ogY0YWuRE4e9celwIthsavas1azJYut0tRwLTJR1ixaxpVQskrQ7shM/h3BVfXLzSaEG3MoUUtBd7nLuXpFa2sCrqqCd56YfsZvYzSlrnOq89LFMGctwHJR+/oZ7MZ+hAXcvibu1op0Pd3ssl6eZztBikDI7lmmZ2RKtOKk1VlVcKOgd4Gp9EPCXp+HXQ16uAjczsuuIkbGhoaBhcJK0K/JeZ/blqWYpC0h54Rbp/4nNOpppZR4vR10UmSZsA140kx74u1FFP8jLIsjc0NCxmkO5lSe8FbrI262DWZo5jQ0NDQ0NDQ0NDQ0NDQz35fxlSsYqU+6axAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\operatorname{Poly}{\\left( \\left(1 - 2 x^{2}\\right) {\\alpha}_{0}{\\alpha}_{1} + \\left(- 2 x^{2} z + z\\right) {\\alpha}_{0} + \\left(- 2 x^{2} y + y\\right) {\\alpha}_{1} -  2 x^{2} y z + y z, {\\alpha}_{0}, {\\alpha}_{1}, domain=\\mathbb{Z}\\left[x, y, z\\right] \\right)}$"
      ],
      "text/plain": [
       "Poly((1 - 2*x**2)*alpha[0]*alpha[1] + (-2*x**2*z + z)*alpha[0] + (-2*x**2*y + y)*alpha[1] - 2*x**2*y*z + y*z, alpha[0], alpha[1], domain='ZZ[x,y,z]')"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "poly = dixon.get_dixon_polynomial()\n",
    "poly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "eeebe728",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 0 & 0 & 1 - 2 x^{2}\\\\0 & 0 & 1 - 2 x^{2} & 0\\\\0 & 1 - 2 x^{2} & 0 & 0\\\\1 - 2 x^{2} & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡                                     2⎤\n",
       "⎢   0         0         0      1 - 2⋅x ⎥\n",
       "⎢                                      ⎥\n",
       "⎢                           2          ⎥\n",
       "⎢   0         0      1 - 2⋅x      0    ⎥\n",
       "⎢                                      ⎥\n",
       "⎢                 2                    ⎥\n",
       "⎢   0      1 - 2⋅x      0         0    ⎥\n",
       "⎢                                      ⎥\n",
       "⎢       2                              ⎥\n",
       "⎣1 - 2⋅x      0         0         0    ⎦"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "matrix = dixon.get_dixon_matrix(poly)\n",
    "matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "7f98bab4",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(- 4 x^{4} + 4 x^{2} - 1\\right)^{2}$"
      ],
      "text/plain": [
       "                   2\n",
       "⎛     4      2    ⎞ \n",
       "⎝- 4⋅x  + 4⋅x  - 1⎠ "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(2 x^{2} - 1\\right)^{4}$"
      ],
      "text/plain": [
       "          4\n",
       "⎛   2    ⎞ \n",
       "⎝2⋅x  - 1⎠ "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ - \\frac{\\sqrt{2}}{2}, \\  \\frac{\\sqrt{2}}{2}\\right]$"
      ],
      "text/plain": [
       "⎡-√2   √2⎤\n",
       "⎢────, ──⎥\n",
       "⎣ 2    2 ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(*[matrix.det(), factor(matrix.det()), solve(factor(matrix.det())) ])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5644efc4",
   "metadata": {},
   "source": [
    "h is in terms of y, z ...can be solved"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "1fac86c8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + y^{2} - 1$"
      ],
      "text/plain": [
       " 2    2    \n",
       "x  + y  - 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( - \\frac{\\sqrt{2}}{2}, \\  \\left[ - \\frac{\\sqrt{2}}{2}, \\  \\frac{\\sqrt{2}}{2}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛-√2   ⎡-√2   √2⎤⎞\n",
       "⎜────, ⎢────, ──⎥⎟\n",
       "⎝ 2    ⎣ 2    2 ⎦⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{\\sqrt{2}}{2}, \\  \\left[ - \\frac{\\sqrt{2}}{2}, \\  \\frac{\\sqrt{2}}{2}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛√2  ⎡-√2   √2⎤⎞\n",
       "⎜──, ⎢────, ──⎥⎟\n",
       "⎝2   ⎣ 2    2 ⎦⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + z^{2} - 1$"
      ],
      "text/plain": [
       " 2    2    \n",
       "x  + z  - 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( - \\frac{\\sqrt{2}}{2}, \\  \\left[ - \\frac{\\sqrt{2}}{2}, \\  \\frac{\\sqrt{2}}{2}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛-√2   ⎡-√2   √2⎤⎞\n",
       "⎜────, ⎢────, ──⎥⎟\n",
       "⎝ 2    ⎣ 2    2 ⎦⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{\\sqrt{2}}{2}, \\  \\left[ - \\frac{\\sqrt{2}}{2}, \\  \\frac{\\sqrt{2}}{2}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛√2  ⎡-√2   √2⎤⎞\n",
       "⎜──, ⎢────, ──⎥⎟\n",
       "⎝2   ⎣ 2    2 ⎦⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle y^{2} + z^{2} - 1$"
      ],
      "text/plain": [
       " 2    2    \n",
       "y  + z  - 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( - \\frac{\\sqrt{2}}{2}, \\  \\left[ \\left\\{ y : - \\sqrt{1 - z^{2}}\\right\\}, \\  \\left\\{ y : \\sqrt{1 - z^{2}}\\right\\}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛      ⎡⎧       ________⎫  ⎧      ________⎫⎤⎞\n",
       "⎜-√2   ⎢⎨      ╱      2 ⎬  ⎨     ╱      2 ⎬⎥⎟\n",
       "⎜────, ⎣⎩y: -╲╱  1 - z  ⎭, ⎩y: ╲╱  1 - z  ⎭⎦⎟\n",
       "⎝ 2                                         ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( \\frac{\\sqrt{2}}{2}, \\  \\left[ \\left\\{ y : - \\sqrt{1 - z^{2}}\\right\\}, \\  \\left\\{ y : \\sqrt{1 - z^{2}}\\right\\}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛    ⎡⎧       ________⎫  ⎧      ________⎫⎤⎞\n",
       "⎜√2  ⎢⎨      ╱      2 ⎬  ⎨     ╱      2 ⎬⎥⎟\n",
       "⎜──, ⎣⎩y: -╲╱  1 - z  ⎭, ⎩y: ╲╱  1 - z  ⎭⎦⎟\n",
       "⎝2                                        ⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for eq in [f,g,h]:\n",
    "    display(eq)\n",
    "    for _x in (-sqrt(2)/2, sqrt(2)/2): #solved x values\n",
    "        y=solve(eq.subs(x,_x))\n",
    "        display((_x, y))"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "20869dc9",
   "metadata": {},
   "source": [
    "# Bezout Resultant $(n=1,2 \\ m=2,3)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "8fde5721",
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy.polys import subresultants_qq_zz\n",
    "var('x');"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8d560fd5",
   "metadata": {},
   "source": [
    "Common root exists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "afa1ecff",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}-1 & 0 & 1\\\\0 & 1 & 1\\\\1 & 1 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡-1  0  1⎤\n",
       "⎢        ⎥\n",
       "⎢0   1  1⎥\n",
       "⎢        ⎥\n",
       "⎣1   1  0⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle 0$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = x ** 3 +1\n",
    "q = x + 1\n",
    "matrix=subresultants_qq_zz.bezout(p, q, x)\n",
    "display(matrix, matrix.det()) #determinant is 0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "d25c0cc8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x755f22f9a0>"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(p,q, xlim=[-5,5], ylim=[-5,5])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ebca6b35",
   "metadata": {},
   "source": [
    "Although we have x3 intersections but only at $x=-1$ we have common root"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "917da683",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\left( -1,\\right)\\right]$"
      ],
      "text/plain": [
       "[(-1,)]"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve((p,q), x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2fca7447",
   "metadata": {},
   "source": [
    "Here we all have intersections, resulting p,q values can be calculated henceforth"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "9296606e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAAAUCAYAAACOPhMlAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjYuMCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy89olMNAAAACXBIWXMAABJ0AAASdAHeZh94AAACiUlEQVR4nO2ZvWsUQRjGfxELUUICilgIiuJHo6YQPwhiJBCCVQpLxdiIKCiincKTR7A0oKJ1QOwVsUmTWCj+B5poIKJoYUQlGgTFs9hZszn2kvvY3YRcfnDM3szuPc/Le8vMvNNSKpVYYWmzarENrLAwq+ML213ASGJsTNLuog01K7Y3AJ+TfZJaIJGkBM+AUWAqByMngKNAB7APaAUeSjqZtVaK9mbgBtALrAc+AY8AS/qag16tsc4ADtf9wJZ4IC1Jo5IGMvJaznUiwz+AD0Ahb6rt7cALYCPwGHgNHAAuAb22OyV9yVi2plglzQADwW8XCyQpTy4TGX5L9C8bmf/2zLhPlKCLku7GnbYHg6ebwLmMNTOLtdAkSfpv1PZ8t2ZGeIt6gEngXrkl4CxwyvYVST+z0s0y1mZY3R0L7bCkv8kBSdPAc2AtcKhoY9XSDEnaFdrxCuNvQruzAC910QxJagvt9wrjcX97/lbqo+o5yfYkiRVHFRSytG4Galk4TAC/arj/Y41e8iJ+U9oqjMf93/K3Uh9VJ0lSd55GcmQstJXmnB2hrTRnLTrNMCfFS+Ee23Pitd0KdBLt9l8WbaxalnySbA/ZLtnur+d5SRPAMLAVuFD+88A64EFyj9SoZtYUupm13Qf0ha+bQnvY9lC4npJ0teyx+I/0pwHp80RloTu2u4FXwEGiPdQ4cC1rzTpjTaXoslAHcLqsb1v4ALwDyo3vAaaBp/WKSpqwvZ/ZAutxogLrbdILrA1rUl+sqRRdFhogFBGrwXY7sBe41WilWtJ74ExRmrXGOh9pSZJtsTTOk44Av4HBZa6Zep4Uk0zSJLPnGZDDeVKtSHoCrFnumoHkedIc/gHQydT5lZkr2QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$\\displaystyle \\left[ -1, \\  0, \\  1\\right]$"
      ],
      "text/plain": [
       "[-1, 0, 1]"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve((p-q),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "91d94751",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x + 1\\right) \\left(x^{2} - x + 1\\right)$"
      ],
      "text/plain": [
       "        ⎛ 2        ⎞\n",
       "(x + 1)⋅⎝x  - x + 1⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x + 1$"
      ],
      "text/plain": [
       "x + 1"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(factor(p), factor(q)) #common root x=-1"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ca3c760b",
   "metadata": {},
   "source": [
    "# Important Case\n",
    "<b> Normal plots contains the points (x,y) where f(x)=y. Implicit plots contains the points (x,y) where f(x,y)=0</b> <br>\n",
    "<font color='red'> <b> Although we have an common intersection but it doesn't translate into common root </b></font> <br> This intersection at $(2,2)$ is <b>a root [or solution] of f(x)-g(x)=0</b> <br> \n",
    "<sub>Thanks to <mark>buster</mark> at https://libera-math.github.io/ for phrasing sentences correctly</sub>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c08d1880",
   "metadata": {},
   "source": [
    "No Common root exists"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "32a95e08",
   "metadata": {
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} - 7 x + 12$"
      ],
      "text/plain": [
       " 2           \n",
       "x  - 7⋅x + 12"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} - x$"
      ],
      "text/plain": [
       " 2    \n",
       "x  - x"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}12 & -12\\\\-12 & 6\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡12   -12⎤\n",
       "⎢        ⎥\n",
       "⎣-12   6 ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle -72$"
      ],
      "text/plain": [
       "-72"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "r = x ** 2 - 7 * x + 12\n",
    "h = x ** 2 - x\n",
    "matrix=subresultants_qq_zz.bezout(r, h, x)\n",
    "display(r, h, matrix, matrix.det()) #determinant is !=0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "f4d07f1e",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x755f157880>"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(r,h, xlim=[-5,5], ylim=[-5,5], markers=[{'args': [2, 2, 'go']} ])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "75ea3887",
   "metadata": {},
   "source": [
    "As expected no common solution"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "063a8e33",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABAAAAAUCAYAAACEYr13AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjYuMCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy89olMNAAAACXBIWXMAABJ0AAASdAHeZh94AAAAgUlEQVR4nO2UMQ7CMAxFXyuOwN61x+AovcmvT0Z6DDZ2OncNAwEcmooha95kxfKLZcnuYozU0FdVA6d3YGYX4OpyN0ljyp2Bhy+U1GUCxwIEYHVvG2ApnoBh14EjSJp/ftuA2XX6EVTPoAmaoAlelLZRZib+3IOS4M535+H4HmQ8ATaYI2kok+bYAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[ \\right]$"
      ],
      "text/plain": [
       "[]"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve((r,h),x)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5878b79d",
   "metadata": {},
   "source": [
    "As mentioned earlier at intersection $r(x)-h(x)=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "495b3c90",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABwAAAAUCAYAAACeXl35AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjYuMCwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy89olMNAAAACXBIWXMAABJ0AAASdAHeZh94AAABd0lEQVR4nL3VvWoUYRTG8V9070DBwkLsLPQeIlYiCCop/bgACQELQZCzjyDYiVhoIcQbsFS00QQ7QRAUUatgY9BgmcKvtcgYh2VjdtfdPM37zpnznP+8w5k5M71ez05q147S0PmzSTKLZ61776vq0LiFk+zFl3asqmY6A3KXsYS1lnkPTuEEjmA/vuE1FrFYVb/66qwjzf4CDtA6YUtLVdXti83hDj7ZeAsfsQ+ncQ/Hk8xV1WZDVNU6us0Dz/4LOEgfcBIP2ydJcgUvcKaBP9iu0FDAqnq6RXw1yV1cx+wwwEl06fdm/TFM8n8Bk3Rwrrl8PHUgbuAwHlXVk6kCk8zjEt7h7LC+sYBJLuIW3uJoVX2dGjDJAm7jTQNbHcU/EjDJZdzEqwb2eRT/SMAkV200yUscq6q1bSwDNdSHn+Q8ruEnnmM+SX/aSlXdnwgQB5t1Nxa2yFnGZIDNz7w7TO44wEpSpjAP+4Er/s4vWvNwTLXn4aZ+A55tdnwmYDxvAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$\\displaystyle \\left[ 2\\right]$"
      ],
      "text/plain": [
       "[2]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve((r-h),x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "40b10226",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left(x - 4\\right) \\left(x - 3\\right)$"
      ],
      "text/plain": [
       "(x - 4)⋅(x - 3)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x \\left(x - 1\\right)$"
      ],
      "text/plain": [
       "x⋅(x - 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "display(factor(r), factor(h)) #no similar factor"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "84959b0a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}x^{2} & 0 & - x\\\\0 & - x & -1\\\\- x & -1 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ 2        ⎤\n",
       "⎢x   0   -x⎥\n",
       "⎢          ⎥\n",
       "⎢0   -x  -1⎥\n",
       "⎢          ⎥\n",
       "⎣-x  -1  0 ⎦"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{3} - x^{2}$"
      ],
      "text/plain": [
       " 3    2\n",
       "x  - x "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} \\left(x - 1\\right)$"
      ],
      "text/plain": [
       " 2        \n",
       "x ⋅(x - 1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var('x,y');\n",
    "p = x + y\n",
    "q = x ** 2 + y ** 3\n",
    "h = x ** 2 + y\n",
    "matrix=subresultants_qq_zz.bezout(p, q, y)\n",
    "display(matrix, matrix.det(), factor(matrix.det())) #determinant factors show x=0 & x=1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "id": "99e7624a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "text/plain": [
       "<sympy.plotting.plot.Plot at 0x755f251100>"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot_implicit(Or(Eq(p,0), Eq(q,0), Eq(h,0)), \\\n",
    "             markers=[{'args': [0, 0, 'go']}, {'args': [1, -1, 'go']} ])"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "44b7d1b9",
   "metadata": {},
   "source": [
    "Two intersections $(0,0) \\ (1,-1)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "id": "95dfb7ca",
   "metadata": {
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x + y$"
      ],
      "text/plain": [
       "x + y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 0, \\  \\left[ 0\\right]\\right)$"
      ],
      "text/plain": [
       "(0, [0])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 1, \\  \\left[ -1\\right]\\right)$"
      ],
      "text/plain": [
       "(1, [-1])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + y^{3}$"
      ],
      "text/plain": [
       " 2    3\n",
       "x  + y "
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 0, \\  \\left[ 0\\right]\\right)$"
      ],
      "text/plain": [
       "(0, [0])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 1, \\  \\left[ -1, \\  \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}, \\  \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right]\\right)$"
      ],
      "text/plain": [
       "⎛   ⎡    1   √3⋅ⅈ  1   √3⋅ⅈ⎤⎞\n",
       "⎜1, ⎢-1, ─ - ────, ─ + ────⎥⎟\n",
       "⎝   ⎣    2    2    2    2  ⎦⎠"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle x^{2} + y$"
      ],
      "text/plain": [
       " 2    \n",
       "x  + y"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 0, \\  \\left[ 0\\right]\\right)$"
      ],
      "text/plain": [
       "(0, [0])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle \\left( 1, \\  \\left[ -1\\right]\\right)$"
      ],
      "text/plain": [
       "(1, [-1])"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for eq in [p,q,h]:\n",
    "    display(eq)\n",
    "    for _x in (0,1): #solved x values\n",
    "        y=solve(eq.subs(x,_x))\n",
    "        display((_x, y))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "id": "bec18d05",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/json": {
       "Software versions": [
        {
         "module": "Python",
         "version": "3.9.13 64bit [Clang 12.0.9 (https://android.googlesource.com/toolchain/llvm-project c935d99d]"
        },
        {
         "module": "IPython",
         "version": "8.4.0"
        },
        {
         "module": "OS",
         "version": "Linux 4.19.113 24856341 aarch64 with libc"
        },
        {
         "module": "sympy",
         "version": "1.11.1"
        }
       ]
      },
      "text/html": [
       "<table><tr><th>Software</th><th>Version</th></tr><tr><td>Python</td><td>3.9.13 64bit [Clang 12.0.9 (https://android.googlesource.com/toolchain/llvm-project c935d99d]</td></tr><tr><td>IPython</td><td>8.4.0</td></tr><tr><td>OS</td><td>Linux 4.19.113 24856341 aarch64 with libc</td></tr><tr><td>sympy</td><td>1.11.1</td></tr><tr><td colspan='2'>Tue Oct 18 06:04:07 2022 PKT</td></tr></table>"
      ],
      "text/latex": [
       "\\begin{tabular}{|l|l|}\\hline\n",
       "{\\bf Software} & {\\bf Version} \\\\ \\hline\\hline\n",
       "Python & 3.9.13 64bit [Clang 12.0.9 (https://android.googlesource.com/toolchain/llvm-project c935d99d] \\\\ \\hline\n",
       "IPython & 8.4.0 \\\\ \\hline\n",
       "OS & Linux 4.19.113 24856341 aarch64 with libc \\\\ \\hline\n",
       "sympy & 1.11.1 \\\\ \\hline\n",
       "\\hline \\multicolumn{2}{|l|}{Tue Oct 18 06:04:07 2022 PKT} \\\\ \\hline\n",
       "\\end{tabular}\n"
      ],
      "text/plain": [
       "Software versions\n",
       "Python 3.9.13 64bit [Clang 12.0.9 (https://android.googlesource.com/toolchain/llvm-project c935d99d]\n",
       "IPython 8.4.0\n",
       "OS Linux 4.19.113 24856341 aarch64 with libc\n",
       "sympy 1.11.1\n",
       "Tue Oct 18 06:04:07 2022 PKT"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%reload_ext version_information\n",
    "%version_information sympy"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "d468d608",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.13"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}