polynomial_substitution_by_nonpower_residue_division.ipynb

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    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
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 },
 "nbformat": 4,
 "nbformat_minor": 2,
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 14,
   "source": [
    "from sympy import Poly,prod,Eq,symbols,Symbol\r\n",
    "from sympy.abc import x \r\n",
    "from IPython.display import display, Latex,HTML"
   ],
   "outputs": [],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "source": [
    "RootDegre=3\r\n",
    "Coeffs=symbols(\"f e d c b a\")"
   ],
   "outputs": [],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "source": [
    "display(Latex(\"$$x \\\\rightarrow x^{}$$\".format(RootDegre)))\r\n",
    "P=Poly.from_list([1] + list(reversed(Coeffs)), gens=x)\r\n",
    "P = P.as_poly(x)\r\n",
    "Eq(P.as_expr(),0)"
   ],
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": [
       "<IPython.core.display.Latex object>"
      ],
      "text/latex": [
       "$$x \\rightarrow x^3$$"
      ]
     },
     "metadata": {}
    },
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Eq(a*x**5 + b*x**4 + c*x**3 + d*x**2 + e*x + f + x**6, 0)"
      ],
      "text/latex": [
       "$\\displaystyle a x^{5} + b x^{4} + c x^{3} + d x^{2} + e x + f + x^{6} = 0$"
      ]
     },
     "metadata": {},
     "execution_count": 16
    }
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "source": [
    "left1=sum([ (b*x**a) for ((a,),b) in  P.all_terms() if a%RootDegre!=0 ]) \r\n",
    "right1=-sum([ (b*x**a) for ((a,),b) in  P.all_terms() if a%RootDegre==0 ]) \r\n",
    "Eq(left1,right1)"
   ],
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Eq(a*x**5 + b*x**4 + d*x**2 + e*x, -c*x**3 - f - x**6)"
      ],
      "text/latex": [
       "$\\displaystyle a x^{5} + b x^{4} + d x^{2} + e x = - c x^{3} - f - x^{6}$"
      ]
     },
     "metadata": {},
     "execution_count": 17
    }
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "source": [
    "left_pol=sum([ (b*x**a) for ((a,),b) in  (left1**RootDegre).as_poly(x).all_terms() if a%RootDegre!=0 ]).as_poly(x)\r\n",
    "left_pol.as_expr()"
   ],
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "3*a**2*b*x**14 + 3*a*b**2*x**13 + 3*d**2*e*x**5 + 3*d*e**2*x**4 + x**11*(3*a**2*e + 6*a*b*d) + x**10*(6*a*b*e + 3*b**2*d) + x**8*(6*a*d*e + 3*b*d**2) + x**7*(3*a*e**2 + 6*b*d*e)"
      ],
      "text/latex": [
       "$\\displaystyle 3 a^{2} b x^{14} + 3 a b^{2} x^{13} + 3 d^{2} e x^{5} + 3 d e^{2} x^{4} + x^{11} \\left(3 a^{2} e + 6 a b d\\right) + x^{10} \\left(6 a b e + 3 b^{2} d\\right) + x^{8} \\left(6 a d e + 3 b d^{2}\\right) + x^{7} \\left(3 a e^{2} + 6 b d e\\right)$"
      ]
     },
     "metadata": {},
     "execution_count": 18
    }
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "source": [
    "common,leftrem = left_pol.div(left1.as_poly(x)) \r\n",
    "if leftrem != 0: \r\n",
    "    #display(leftrem.as_expr())\r\n",
    "    raise SystemExit(\"Not exists!\")\r\n",
    "Eq(common.as_expr()*left1,common.as_expr()*right1)"
   ],
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Eq((3*a*b*x**9 + 3*d*e*x**3 + x**6*(3*a*e + 3*b*d))*(a*x**5 + b*x**4 + d*x**2 + e*x), (-c*x**3 - f - x**6)*(3*a*b*x**9 + 3*d*e*x**3 + x**6*(3*a*e + 3*b*d)))"
      ],
      "text/latex": [
       "$\\displaystyle \\left(3 a b x^{9} + 3 d e x^{3} + x^{6} \\left(3 a e + 3 b d\\right)\\right) \\left(a x^{5} + b x^{4} + d x^{2} + e x\\right) = \\left(- c x^{3} - f - x^{6}\\right) \\left(3 a b x^{9} + 3 d e x^{3} + x^{6} \\left(3 a e + 3 b d\\right)\\right)$"
      ]
     },
     "metadata": {},
     "execution_count": 19
    }
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "source": [
    "Eq((common.as_expr()*left1).expand(),common.as_expr()*right1)"
   ],
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Eq(3*a**2*b*x**14 + 3*a**2*e*x**11 + 3*a*b**2*x**13 + 6*a*b*d*x**11 + 6*a*b*e*x**10 + 6*a*d*e*x**8 + 3*a*e**2*x**7 + 3*b**2*d*x**10 + 3*b*d**2*x**8 + 6*b*d*e*x**7 + 3*d**2*e*x**5 + 3*d*e**2*x**4, (-c*x**3 - f - x**6)*(3*a*b*x**9 + 3*d*e*x**3 + x**6*(3*a*e + 3*b*d)))"
      ],
      "text/latex": [
       "$\\displaystyle 3 a^{2} b x^{14} + 3 a^{2} e x^{11} + 3 a b^{2} x^{13} + 6 a b d x^{11} + 6 a b e x^{10} + 6 a d e x^{8} + 3 a e^{2} x^{7} + 3 b^{2} d x^{10} + 3 b d^{2} x^{8} + 6 b d e x^{7} + 3 d^{2} e x^{5} + 3 d e^{2} x^{4} = \\left(- c x^{3} - f - x^{6}\\right) \\left(3 a b x^{9} + 3 d e x^{3} + x^{6} \\left(3 a e + 3 b d\\right)\\right)$"
      ]
     },
     "metadata": {},
     "execution_count": 20
    }
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "source": [
    "left2 = left1**RootDegre\r\n",
    "right2 = right1**RootDegre\r\n",
    "Eq(left2,right2)"
   ],
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Eq((a*x**5 + b*x**4 + d*x**2 + e*x)**3, (-c*x**3 - f - x**6)**3)"
      ],
      "text/latex": [
       "$\\displaystyle \\left(a x^{5} + b x^{4} + d x^{2} + e x\\right)^{3} = \\left(- c x^{3} - f - x^{6}\\right)^{3}$"
      ]
     },
     "metadata": {},
     "execution_count": 21
    }
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "source": [
    "Eq(left2.expand(),right2)"
   ],
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Eq(a**3*x**15 + 3*a**2*b*x**14 + 3*a**2*d*x**12 + 3*a**2*e*x**11 + 3*a*b**2*x**13 + 6*a*b*d*x**11 + 6*a*b*e*x**10 + 3*a*d**2*x**9 + 6*a*d*e*x**8 + 3*a*e**2*x**7 + b**3*x**12 + 3*b**2*d*x**10 + 3*b**2*e*x**9 + 3*b*d**2*x**8 + 6*b*d*e*x**7 + 3*b*e**2*x**6 + d**3*x**6 + 3*d**2*e*x**5 + 3*d*e**2*x**4 + e**3*x**3, (-c*x**3 - f - x**6)**3)"
      ],
      "text/latex": [
       "$\\displaystyle a^{3} x^{15} + 3 a^{2} b x^{14} + 3 a^{2} d x^{12} + 3 a^{2} e x^{11} + 3 a b^{2} x^{13} + 6 a b d x^{11} + 6 a b e x^{10} + 3 a d^{2} x^{9} + 6 a d e x^{8} + 3 a e^{2} x^{7} + b^{3} x^{12} + 3 b^{2} d x^{10} + 3 b^{2} e x^{9} + 3 b d^{2} x^{8} + 6 b d e x^{7} + 3 b e^{2} x^{6} + d^{3} x^{6} + 3 d^{2} e x^{5} + 3 d e^{2} x^{4} + e^{3} x^{3} = \\left(- c x^{3} - f - x^{6}\\right)^{3}$"
      ]
     },
     "metadata": {},
     "execution_count": 22
    }
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "source": [
    "left2 = left1**RootDegre  -  common.as_expr()*left1\r\n",
    "right2 = right1**RootDegre - common.as_expr()*right1\r\n",
    "Eq(left2.expand(),right2)"
   ],
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Eq(a**3*x**15 + 3*a**2*d*x**12 + 3*a*d**2*x**9 + b**3*x**12 + 3*b**2*e*x**9 + 3*b*e**2*x**6 + d**3*x**6 + e**3*x**3, (-c*x**3 - f - x**6)**3 - (-c*x**3 - f - x**6)*(3*a*b*x**9 + 3*d*e*x**3 + x**6*(3*a*e + 3*b*d)))"
      ],
      "text/latex": [
       "$\\displaystyle a^{3} x^{15} + 3 a^{2} d x^{12} + 3 a d^{2} x^{9} + b^{3} x^{12} + 3 b^{2} e x^{9} + 3 b e^{2} x^{6} + d^{3} x^{6} + e^{3} x^{3} = \\left(- c x^{3} - f - x^{6}\\right)^{3} - \\left(- c x^{3} - f - x^{6}\\right) \\left(3 a b x^{9} + 3 d e x^{3} + x^{6} \\left(3 a e + 3 b d\\right)\\right)$"
      ]
     },
     "metadata": {},
     "execution_count": 23
    }
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "source": [
    "P2 = (left2-right2).as_poly(x)\r\n",
    "Eq(P2.as_expr(),0)"
   ],
   "outputs": [
    {
     "output_type": "execute_result",
     "data": {
      "text/plain": [
       "Eq(f**3 + x**18 + x**15*(a**3 - 3*a*b + 3*c) + x**12*(3*a**2*d - 3*a*b*c - 3*a*e + b**3 - 3*b*d + 3*c**2 + 3*f) + x**9*(-3*a*b*f - 3*a*c*e + 3*a*d**2 + 3*b**2*e - 3*b*c*d + c**3 + 6*c*f - 3*d*e) + x**6*(-3*a*e*f - 3*b*d*f + 3*b*e**2 + 3*c**2*f - 3*c*d*e + d**3 + 3*f**2) + x**3*(3*c*f**2 - 3*d*e*f + e**3), 0)"
      ],
      "text/latex": [
       "$\\displaystyle f^{3} + x^{18} + x^{15} \\left(a^{3} - 3 a b + 3 c\\right) + x^{12} \\left(3 a^{2} d - 3 a b c - 3 a e + b^{3} - 3 b d + 3 c^{2} + 3 f\\right) + x^{9} \\left(- 3 a b f - 3 a c e + 3 a d^{2} + 3 b^{2} e - 3 b c d + c^{3} + 6 c f - 3 d e\\right) + x^{6} \\left(- 3 a e f - 3 b d f + 3 b e^{2} + 3 c^{2} f - 3 c d e + d^{3} + 3 f^{2}\\right) + x^{3} \\left(3 c f^{2} - 3 d e f + e^{3}\\right) = 0$"
      ]
     },
     "metadata": {},
     "execution_count": 24
    }
   ],
   "metadata": {}
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "source": [
    "for a,b in zip(Coeffs[::-1],P2.terms()[1:]):\r\n",
    "    display(Eq(symbols(a.name.upper()),b[1]))"
   ],
   "outputs": [
    {
     "output_type": "display_data",
     "data": {
      "text/plain": [
       "Eq(A, a**3 - 3*a*b + 3*c)"
      ],
      "text/latex": [
       "$\\displaystyle A = a^{3} - 3 a b + 3 c$"
      ]
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": [
       "Eq(B, 3*a**2*d - 3*a*b*c - 3*a*e + b**3 - 3*b*d + 3*c**2 + 3*f)"
      ],
      "text/latex": [
       "$\\displaystyle B = 3 a^{2} d - 3 a b c - 3 a e + b^{3} - 3 b d + 3 c^{2} + 3 f$"
      ]
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": [
       "Eq(C, -3*a*b*f - 3*a*c*e + 3*a*d**2 + 3*b**2*e - 3*b*c*d + c**3 + 6*c*f - 3*d*e)"
      ],
      "text/latex": [
       "$\\displaystyle C = - 3 a b f - 3 a c e + 3 a d^{2} + 3 b^{2} e - 3 b c d + c^{3} + 6 c f - 3 d e$"
      ]
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": [
       "Eq(D, -3*a*e*f - 3*b*d*f + 3*b*e**2 + 3*c**2*f - 3*c*d*e + d**3 + 3*f**2)"
      ],
      "text/latex": [
       "$\\displaystyle D = - 3 a e f - 3 b d f + 3 b e^{2} + 3 c^{2} f - 3 c d e + d^{3} + 3 f^{2}$"
      ]
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": [
       "Eq(E, 3*c*f**2 - 3*d*e*f + e**3)"
      ],
      "text/latex": [
       "$\\displaystyle E = 3 c f^{2} - 3 d e f + e^{3}$"
      ]
     },
     "metadata": {}
    },
    {
     "output_type": "display_data",
     "data": {
      "text/plain": [
       "Eq(F, f**3)"
      ],
      "text/latex": [
       "$\\displaystyle F = f^{3}$"
      ]
     },
     "metadata": {}
    }
   ],
   "metadata": {}
  }
 ]
}

polynomial_vieta_power_substitution.ipynb