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ArpegiusWhooves/polynomial_substitution_by_nonpower_residue_division.ipynb

Last active July 26, 2021 11:44
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polynomial_vieta_power_substitution.ipynb
{
"cells": [
{
"cell_type": "code",
"execution_count": 100,
"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\r\n",
"from sympy.polys.polyfuncs import symmetrize\r\n",
"from sympy.polys.specialpolys import symmetric_poly"
],
"outputs": [],
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 101,
"source": [
"RootDegre=5\r\n",
"Coeffs=symbols(\"f e d c b a\")"
],
"outputs": [],
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 102,
"source": [
"R=symbols(f\"r:{len(Coeffs)}\")\r\n",
"P=prod([(x-r) for r in R]).as_poly(x)\r\n",
"display(Eq(P.as_expr(),0)) \r\n",
"display(Eq(Poly.from_list((1,)+Coeffs[::-1] ,x).as_expr(),0))\r\n",
"AL=[]\r\n",
"for (d,),v in P.terms():\r\n",
" if v != 1:\r\n",
" display(Eq(Coeffs[d],v))\r\n",
" AL.append((Coeffs[d],v))"
],
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(r0*r1*r2*r3*r4*r5 + x**6 + x**5*(-r0 - r1 - r2 - r3 - r4 - r5) + x**4*(r0*r1 + r0*r2 + r0*r3 + r0*r4 + r0*r5 + r1*r2 + r1*r3 + r1*r4 + r1*r5 + r2*r3 + r2*r4 + r2*r5 + r3*r4 + r3*r5 + r4*r5) + x**3*(-r0*r1*r2 - r0*r1*r3 - r0*r1*r4 - r0*r1*r5 - r0*r2*r3 - r0*r2*r4 - r0*r2*r5 - r0*r3*r4 - r0*r3*r5 - r0*r4*r5 - r1*r2*r3 - r1*r2*r4 - r1*r2*r5 - r1*r3*r4 - r1*r3*r5 - r1*r4*r5 - r2*r3*r4 - r2*r3*r5 - r2*r4*r5 - r3*r4*r5) + x**2*(r0*r1*r2*r3 + r0*r1*r2*r4 + r0*r1*r2*r5 + r0*r1*r3*r4 + r0*r1*r3*r5 + r0*r1*r4*r5 + r0*r2*r3*r4 + r0*r2*r3*r5 + r0*r2*r4*r5 + r0*r3*r4*r5 + r1*r2*r3*r4 + r1*r2*r3*r5 + r1*r2*r4*r5 + r1*r3*r4*r5 + r2*r3*r4*r5) + x*(-r0*r1*r2*r3*r4 - r0*r1*r2*r3*r5 - r0*r1*r2*r4*r5 - r0*r1*r3*r4*r5 - r0*r2*r3*r4*r5 - r1*r2*r3*r4*r5), 0)"
],
"text/latex": [
"$\\displaystyle r_{0} r_{1} r_{2} r_{3} r_{4} r_{5} + x^{6} + x^{5} \\left(- r_{0} - r_{1} - r_{2} - r_{3} - r_{4} - r_{5}\\right) + x^{4} \\left(r_{0} r_{1} + r_{0} r_{2} + r_{0} r_{3} + r_{0} r_{4} + r_{0} r_{5} + r_{1} r_{2} + r_{1} r_{3} + r_{1} r_{4} + r_{1} r_{5} + r_{2} r_{3} + r_{2} r_{4} + r_{2} r_{5} + r_{3} r_{4} + r_{3} r_{5} + r_{4} r_{5}\\right) + x^{3} \\left(- r_{0} r_{1} r_{2} - r_{0} r_{1} r_{3} - r_{0} r_{1} r_{4} - r_{0} r_{1} r_{5} - r_{0} r_{2} r_{3} - r_{0} r_{2} r_{4} - r_{0} r_{2} r_{5} - r_{0} r_{3} r_{4} - r_{0} r_{3} r_{5} - r_{0} r_{4} r_{5} - r_{1} r_{2} r_{3} - r_{1} r_{2} r_{4} - r_{1} r_{2} r_{5} - r_{1} r_{3} r_{4} - r_{1} r_{3} r_{5} - r_{1} r_{4} r_{5} - r_{2} r_{3} r_{4} - r_{2} r_{3} r_{5} - r_{2} r_{4} r_{5} - r_{3} r_{4} r_{5}\\right) + x^{2} \\left(r_{0} r_{1} r_{2} r_{3} + r_{0} r_{1} r_{2} r_{4} + r_{0} r_{1} r_{2} r_{5} + r_{0} r_{1} r_{3} r_{4} + r_{0} r_{1} r_{3} r_{5} + r_{0} r_{1} r_{4} r_{5} + r_{0} r_{2} r_{3} r_{4} + r_{0} r_{2} r_{3} r_{5} + r_{0} r_{2} r_{4} r_{5} + r_{0} r_{3} r_{4} r_{5} + r_{1} r_{2} r_{3} r_{4} + r_{1} r_{2} r_{3} r_{5} + r_{1} r_{2} r_{4} r_{5} + r_{1} r_{3} r_{4} r_{5} + r_{2} r_{3} r_{4} r_{5}\\right) + x \\left(- r_{0} r_{1} r_{2} r_{3} r_{4} - r_{0} r_{1} r_{2} r_{3} r_{5} - r_{0} r_{1} r_{2} r_{4} r_{5} - r_{0} r_{1} r_{3} r_{4} r_{5} - r_{0} r_{2} r_{3} r_{4} r_{5} - r_{1} r_{2} r_{3} r_{4} r_{5}\\right) = 0$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"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": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(a, -r0 - r1 - r2 - r3 - r4 - r5)"
],
"text/latex": [
"$\\displaystyle a = - r_{0} - r_{1} - r_{2} - r_{3} - r_{4} - r_{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(b, r0*r1 + r0*r2 + r0*r3 + r0*r4 + r0*r5 + r1*r2 + r1*r3 + r1*r4 + r1*r5 + r2*r3 + r2*r4 + r2*r5 + r3*r4 + r3*r5 + r4*r5)"
],
"text/latex": [
"$\\displaystyle b = r_{0} r_{1} + r_{0} r_{2} + r_{0} r_{3} + r_{0} r_{4} + r_{0} r_{5} + r_{1} r_{2} + r_{1} r_{3} + r_{1} r_{4} + r_{1} r_{5} + r_{2} r_{3} + r_{2} r_{4} + r_{2} r_{5} + r_{3} r_{4} + r_{3} r_{5} + r_{4} r_{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(c, -r0*r1*r2 - r0*r1*r3 - r0*r1*r4 - r0*r1*r5 - r0*r2*r3 - r0*r2*r4 - r0*r2*r5 - r0*r3*r4 - r0*r3*r5 - r0*r4*r5 - r1*r2*r3 - r1*r2*r4 - r1*r2*r5 - r1*r3*r4 - r1*r3*r5 - r1*r4*r5 - r2*r3*r4 - r2*r3*r5 - r2*r4*r5 - r3*r4*r5)"
],
"text/latex": [
"$\\displaystyle c = - r_{0} r_{1} r_{2} - r_{0} r_{1} r_{3} - r_{0} r_{1} r_{4} - r_{0} r_{1} r_{5} - r_{0} r_{2} r_{3} - r_{0} r_{2} r_{4} - r_{0} r_{2} r_{5} - r_{0} r_{3} r_{4} - r_{0} r_{3} r_{5} - r_{0} r_{4} r_{5} - r_{1} r_{2} r_{3} - r_{1} r_{2} r_{4} - r_{1} r_{2} r_{5} - r_{1} r_{3} r_{4} - r_{1} r_{3} r_{5} - r_{1} r_{4} r_{5} - r_{2} r_{3} r_{4} - r_{2} r_{3} r_{5} - r_{2} r_{4} r_{5} - r_{3} r_{4} r_{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(d, r0*r1*r2*r3 + r0*r1*r2*r4 + r0*r1*r2*r5 + r0*r1*r3*r4 + r0*r1*r3*r5 + r0*r1*r4*r5 + r0*r2*r3*r4 + r0*r2*r3*r5 + r0*r2*r4*r5 + r0*r3*r4*r5 + r1*r2*r3*r4 + r1*r2*r3*r5 + r1*r2*r4*r5 + r1*r3*r4*r5 + r2*r3*r4*r5)"
],
"text/latex": [
"$\\displaystyle d = r_{0} r_{1} r_{2} r_{3} + r_{0} r_{1} r_{2} r_{4} + r_{0} r_{1} r_{2} r_{5} + r_{0} r_{1} r_{3} r_{4} + r_{0} r_{1} r_{3} r_{5} + r_{0} r_{1} r_{4} r_{5} + r_{0} r_{2} r_{3} r_{4} + r_{0} r_{2} r_{3} r_{5} + r_{0} r_{2} r_{4} r_{5} + r_{0} r_{3} r_{4} r_{5} + r_{1} r_{2} r_{3} r_{4} + r_{1} r_{2} r_{3} r_{5} + r_{1} r_{2} r_{4} r_{5} + r_{1} r_{3} r_{4} r_{5} + r_{2} r_{3} r_{4} r_{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(e, -r0*r1*r2*r3*r4 - r0*r1*r2*r3*r5 - r0*r1*r2*r4*r5 - r0*r1*r3*r4*r5 - r0*r2*r3*r4*r5 - r1*r2*r3*r4*r5)"
],
"text/latex": [
"$\\displaystyle e = - r_{0} r_{1} r_{2} r_{3} r_{4} - r_{0} r_{1} r_{2} r_{3} r_{5} - r_{0} r_{1} r_{2} r_{4} r_{5} - r_{0} r_{1} r_{3} r_{4} r_{5} - r_{0} r_{2} r_{3} r_{4} r_{5} - r_{1} r_{2} r_{3} r_{4} r_{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(f, r0*r1*r2*r3*r4*r5)"
],
"text/latex": [
"$\\displaystyle f = r_{0} r_{1} r_{2} r_{3} r_{4} r_{5}$"
]
},
"metadata": {}
}
],
"metadata": {}
},
{
"cell_type": "code",
"execution_count": 103,
"source": [
"P=prod([(x**RootDegre-r**RootDegre) for r in R]).as_poly(x)\r\n",
"display(Eq(P.as_expr(),0))\r\n",
"AA=[ symbols(c.name.upper()) for c in Coeffs ]\r\n",
"display(Eq(Poly.from_list([1]+AA[::-1] ,x).as_expr(),0))\r\n",
"for d,v in P.terms()[1:]:\r\n",
" d=d[0]//RootDegre\r\n",
" mp=(-1)**(P.degree()-d)\r\n",
" V=symmetrize(v,formal=True,symbols=Coeffs[::-1])[0]*mp\r\n",
" display(Eq(AA[d],v,evaluate=False),Eq(AA[d],V,evaluate=False))\r\n",
" #assert(v==V.subs(AL).expand())"
],
"outputs": [
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(r0**5*r1**5*r2**5*r3**5*r4**5*r5**5 + x**30 + x**25*(-r0**5 - r1**5 - r2**5 - r3**5 - r4**5 - r5**5) + x**20*(r0**5*r1**5 + r0**5*r2**5 + r0**5*r3**5 + r0**5*r4**5 + r0**5*r5**5 + r1**5*r2**5 + r1**5*r3**5 + r1**5*r4**5 + r1**5*r5**5 + r2**5*r3**5 + r2**5*r4**5 + r2**5*r5**5 + r3**5*r4**5 + r3**5*r5**5 + r4**5*r5**5) + x**15*(-r0**5*r1**5*r2**5 - r0**5*r1**5*r3**5 - r0**5*r1**5*r4**5 - r0**5*r1**5*r5**5 - r0**5*r2**5*r3**5 - r0**5*r2**5*r4**5 - r0**5*r2**5*r5**5 - r0**5*r3**5*r4**5 - r0**5*r3**5*r5**5 - r0**5*r4**5*r5**5 - r1**5*r2**5*r3**5 - r1**5*r2**5*r4**5 - r1**5*r2**5*r5**5 - r1**5*r3**5*r4**5 - r1**5*r3**5*r5**5 - r1**5*r4**5*r5**5 - r2**5*r3**5*r4**5 - r2**5*r3**5*r5**5 - r2**5*r4**5*r5**5 - r3**5*r4**5*r5**5) + x**10*(r0**5*r1**5*r2**5*r3**5 + r0**5*r1**5*r2**5*r4**5 + r0**5*r1**5*r2**5*r5**5 + r0**5*r1**5*r3**5*r4**5 + r0**5*r1**5*r3**5*r5**5 + r0**5*r1**5*r4**5*r5**5 + r0**5*r2**5*r3**5*r4**5 + r0**5*r2**5*r3**5*r5**5 + r0**5*r2**5*r4**5*r5**5 + r0**5*r3**5*r4**5*r5**5 + r1**5*r2**5*r3**5*r4**5 + r1**5*r2**5*r3**5*r5**5 + r1**5*r2**5*r4**5*r5**5 + r1**5*r3**5*r4**5*r5**5 + r2**5*r3**5*r4**5*r5**5) + x**5*(-r0**5*r1**5*r2**5*r3**5*r4**5 - r0**5*r1**5*r2**5*r3**5*r5**5 - r0**5*r1**5*r2**5*r4**5*r5**5 - r0**5*r1**5*r3**5*r4**5*r5**5 - r0**5*r2**5*r3**5*r4**5*r5**5 - r1**5*r2**5*r3**5*r4**5*r5**5), 0)"
],
"text/latex": [
"$\\displaystyle r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5} + x^{30} + x^{25} \\left(- r_{0}^{5} - r_{1}^{5} - r_{2}^{5} - r_{3}^{5} - r_{4}^{5} - r_{5}^{5}\\right) + x^{20} \\left(r_{0}^{5} r_{1}^{5} + r_{0}^{5} r_{2}^{5} + r_{0}^{5} r_{3}^{5} + r_{0}^{5} r_{4}^{5} + r_{0}^{5} r_{5}^{5} + r_{1}^{5} r_{2}^{5} + r_{1}^{5} r_{3}^{5} + r_{1}^{5} r_{4}^{5} + r_{1}^{5} r_{5}^{5} + r_{2}^{5} r_{3}^{5} + r_{2}^{5} r_{4}^{5} + r_{2}^{5} r_{5}^{5} + r_{3}^{5} r_{4}^{5} + r_{3}^{5} r_{5}^{5} + r_{4}^{5} r_{5}^{5}\\right) + x^{15} \\left(- r_{0}^{5} r_{1}^{5} r_{2}^{5} - r_{0}^{5} r_{1}^{5} r_{3}^{5} - r_{0}^{5} r_{1}^{5} r_{4}^{5} - r_{0}^{5} r_{1}^{5} r_{5}^{5} - r_{0}^{5} r_{2}^{5} r_{3}^{5} - r_{0}^{5} r_{2}^{5} r_{4}^{5} - r_{0}^{5} r_{2}^{5} r_{5}^{5} - r_{0}^{5} r_{3}^{5} r_{4}^{5} - r_{0}^{5} r_{3}^{5} r_{5}^{5} - r_{0}^{5} r_{4}^{5} r_{5}^{5} - r_{1}^{5} r_{2}^{5} r_{3}^{5} - r_{1}^{5} r_{2}^{5} r_{4}^{5} - r_{1}^{5} r_{2}^{5} r_{5}^{5} - r_{1}^{5} r_{3}^{5} r_{4}^{5} - r_{1}^{5} r_{3}^{5} r_{5}^{5} - r_{1}^{5} r_{4}^{5} r_{5}^{5} - r_{2}^{5} r_{3}^{5} r_{4}^{5} - r_{2}^{5} r_{3}^{5} r_{5}^{5} - r_{2}^{5} r_{4}^{5} r_{5}^{5} - r_{3}^{5} r_{4}^{5} r_{5}^{5}\\right) + x^{10} \\left(r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{3}^{5} + r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{4}^{5} + r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{5}^{5} + r_{0}^{5} r_{1}^{5} r_{3}^{5} r_{4}^{5} + r_{0}^{5} r_{1}^{5} r_{3}^{5} r_{5}^{5} + r_{0}^{5} r_{1}^{5} r_{4}^{5} r_{5}^{5} + r_{0}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} + r_{0}^{5} r_{2}^{5} r_{3}^{5} r_{5}^{5} + r_{0}^{5} r_{2}^{5} r_{4}^{5} r_{5}^{5} + r_{0}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5} + r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} + r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{5}^{5} + r_{1}^{5} r_{2}^{5} r_{4}^{5} r_{5}^{5} + r_{1}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5} + r_{2}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5}\\right) + x^{5} \\left(- r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} - r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{5}^{5} - r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{4}^{5} r_{5}^{5} - r_{0}^{5} r_{1}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5} - r_{0}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5} - r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5}\\right) = 0$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"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": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(A, -r0**5 - r1**5 - r2**5 - r3**5 - r4**5 - r5**5)"
],
"text/latex": [
"$\\displaystyle A = - r_{0}^{5} - r_{1}^{5} - r_{2}^{5} - r_{3}^{5} - r_{4}^{5} - r_{5}^{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(A, a**5 - 5*a**3*b + 5*a**2*c + 5*a*b**2 - 5*a*d - 5*b*c + 5*e)"
],
"text/latex": [
"$\\displaystyle A = a^{5} - 5 a^{3} b + 5 a^{2} c + 5 a b^{2} - 5 a d - 5 b c + 5 e$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(B, r0**5*r1**5 + r0**5*r2**5 + r0**5*r3**5 + r0**5*r4**5 + r0**5*r5**5 + r1**5*r2**5 + r1**5*r3**5 + r1**5*r4**5 + r1**5*r5**5 + r2**5*r3**5 + r2**5*r4**5 + r2**5*r5**5 + r3**5*r4**5 + r3**5*r5**5 + r4**5*r5**5)"
],
"text/latex": [
"$\\displaystyle B = r_{0}^{5} r_{1}^{5} + r_{0}^{5} r_{2}^{5} + r_{0}^{5} r_{3}^{5} + r_{0}^{5} r_{4}^{5} + r_{0}^{5} r_{5}^{5} + r_{1}^{5} r_{2}^{5} + r_{1}^{5} r_{3}^{5} + r_{1}^{5} r_{4}^{5} + r_{1}^{5} r_{5}^{5} + r_{2}^{5} r_{3}^{5} + r_{2}^{5} r_{4}^{5} + r_{2}^{5} r_{5}^{5} + r_{3}^{5} r_{4}^{5} + r_{3}^{5} r_{5}^{5} + r_{4}^{5} r_{5}^{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(B, 5*a**4*f - 5*a**3*b*e - 5*a**3*c*d + 5*a**2*b**2*d + 5*a**2*b*c**2 - 15*a**2*b*f + 10*a**2*c*e + 5*a**2*d**2 - 5*a*b**3*c + 10*a*b**2*e - 5*a*b*c*d - 5*a*c**3 + 10*a*c*f - 15*a*d*e + b**5 - 5*b**3*d + 5*b**2*c**2 + 5*b**2*f - 15*b*c*e + 5*b*d**2 + 5*c**2*d - 5*d*f + 10*e**2)"
],
"text/latex": [
"$\\displaystyle B = 5 a^{4} f - 5 a^{3} b e - 5 a^{3} c d + 5 a^{2} b^{2} d + 5 a^{2} b c^{2} - 15 a^{2} b f + 10 a^{2} c e + 5 a^{2} d^{2} - 5 a b^{3} c + 10 a b^{2} e - 5 a b c d - 5 a c^{3} + 10 a c f - 15 a d e + b^{5} - 5 b^{3} d + 5 b^{2} c^{2} + 5 b^{2} f - 15 b c e + 5 b d^{2} + 5 c^{2} d - 5 d f + 10 e^{2}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(C, -r0**5*r1**5*r2**5 - r0**5*r1**5*r3**5 - r0**5*r1**5*r4**5 - r0**5*r1**5*r5**5 - r0**5*r2**5*r3**5 - r0**5*r2**5*r4**5 - r0**5*r2**5*r5**5 - r0**5*r3**5*r4**5 - r0**5*r3**5*r5**5 - r0**5*r4**5*r5**5 - r1**5*r2**5*r3**5 - r1**5*r2**5*r4**5 - r1**5*r2**5*r5**5 - r1**5*r3**5*r4**5 - r1**5*r3**5*r5**5 - r1**5*r4**5*r5**5 - r2**5*r3**5*r4**5 - r2**5*r3**5*r5**5 - r2**5*r4**5*r5**5 - r3**5*r4**5*r5**5)"
],
"text/latex": [
"$\\displaystyle C = - r_{0}^{5} r_{1}^{5} r_{2}^{5} - r_{0}^{5} r_{1}^{5} r_{3}^{5} - r_{0}^{5} r_{1}^{5} r_{4}^{5} - r_{0}^{5} r_{1}^{5} r_{5}^{5} - r_{0}^{5} r_{2}^{5} r_{3}^{5} - r_{0}^{5} r_{2}^{5} r_{4}^{5} - r_{0}^{5} r_{2}^{5} r_{5}^{5} - r_{0}^{5} r_{3}^{5} r_{4}^{5} - r_{0}^{5} r_{3}^{5} r_{5}^{5} - r_{0}^{5} r_{4}^{5} r_{5}^{5} - r_{1}^{5} r_{2}^{5} r_{3}^{5} - r_{1}^{5} r_{2}^{5} r_{4}^{5} - r_{1}^{5} r_{2}^{5} r_{5}^{5} - r_{1}^{5} r_{3}^{5} r_{4}^{5} - r_{1}^{5} r_{3}^{5} r_{5}^{5} - r_{1}^{5} r_{4}^{5} r_{5}^{5} - r_{2}^{5} r_{3}^{5} r_{4}^{5} - r_{2}^{5} r_{3}^{5} r_{5}^{5} - r_{2}^{5} r_{4}^{5} r_{5}^{5} - r_{3}^{5} r_{4}^{5} r_{5}^{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(C, 10*a**3*f**2 - 15*a**2*b*e*f - 15*a**2*c*d*f + 5*a**2*c*e**2 + 5*a**2*d**2*e + 10*a*b**2*d*f + 5*a*b**2*e**2 + 10*a*b*c**2*f - 5*a*b*c*d*e - 5*a*b*d**3 - 15*a*b*f**2 - 5*a*c**3*e + 5*a*c**2*d**2 + 20*a*c*e*f + 10*a*d**2*f - 15*a*d*e**2 - 5*b**3*c*f - 5*b**3*d*e + 5*b**2*c**2*e + 5*b**2*c*d**2 + 10*b**2*e*f - 5*b*c**3*d - 5*b*c*d*f - 15*b*c*e**2 + 10*b*d**2*e + c**5 - 5*c**3*f + 10*c**2*d*e - 5*c*d**3 + 5*c*f**2 - 15*d*e*f + 10*e**3)"
],
"text/latex": [
"$\\displaystyle C = 10 a^{3} f^{2} - 15 a^{2} b e f - 15 a^{2} c d f + 5 a^{2} c e^{2} + 5 a^{2} d^{2} e + 10 a b^{2} d f + 5 a b^{2} e^{2} + 10 a b c^{2} f - 5 a b c d e - 5 a b d^{3} - 15 a b f^{2} - 5 a c^{3} e + 5 a c^{2} d^{2} + 20 a c e f + 10 a d^{2} f - 15 a d e^{2} - 5 b^{3} c f - 5 b^{3} d e + 5 b^{2} c^{2} e + 5 b^{2} c d^{2} + 10 b^{2} e f - 5 b c^{3} d - 5 b c d f - 15 b c e^{2} + 10 b d^{2} e + c^{5} - 5 c^{3} f + 10 c^{2} d e - 5 c d^{3} + 5 c f^{2} - 15 d e f + 10 e^{3}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(D, r0**5*r1**5*r2**5*r3**5 + r0**5*r1**5*r2**5*r4**5 + r0**5*r1**5*r2**5*r5**5 + r0**5*r1**5*r3**5*r4**5 + r0**5*r1**5*r3**5*r5**5 + r0**5*r1**5*r4**5*r5**5 + r0**5*r2**5*r3**5*r4**5 + r0**5*r2**5*r3**5*r5**5 + r0**5*r2**5*r4**5*r5**5 + r0**5*r3**5*r4**5*r5**5 + r1**5*r2**5*r3**5*r4**5 + r1**5*r2**5*r3**5*r5**5 + r1**5*r2**5*r4**5*r5**5 + r1**5*r3**5*r4**5*r5**5 + r2**5*r3**5*r4**5*r5**5)"
],
"text/latex": [
"$\\displaystyle D = r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{3}^{5} + r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{4}^{5} + r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{5}^{5} + r_{0}^{5} r_{1}^{5} r_{3}^{5} r_{4}^{5} + r_{0}^{5} r_{1}^{5} r_{3}^{5} r_{5}^{5} + r_{0}^{5} r_{1}^{5} r_{4}^{5} r_{5}^{5} + r_{0}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} + r_{0}^{5} r_{2}^{5} r_{3}^{5} r_{5}^{5} + r_{0}^{5} r_{2}^{5} r_{4}^{5} r_{5}^{5} + r_{0}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5} + r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} + r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{5}^{5} + r_{1}^{5} r_{2}^{5} r_{4}^{5} r_{5}^{5} + r_{1}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5} + r_{2}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(D, 10*a**2*f**3 - 15*a*b*e*f**2 - 15*a*c*d*f**2 + 10*a*c*e**2*f + 10*a*d**2*e*f - 5*a*d*e**3 + 5*b**2*d*f**2 + 5*b**2*e**2*f + 5*b*c**2*f**2 - 5*b*c*d*e*f - 5*b*c*e**3 - 5*b*d**3*f + 5*b*d**2*e**2 - 5*b*f**3 - 5*c**3*e*f + 5*c**2*d**2*f + 5*c**2*d*e**2 - 5*c*d**3*e + 10*c*e*f**2 + d**5 + 5*d**2*f**2 - 15*d*e**2*f + 5*e**4)"
],
"text/latex": [
"$\\displaystyle D = 10 a^{2} f^{3} - 15 a b e f^{2} - 15 a c d f^{2} + 10 a c e^{2} f + 10 a d^{2} e f - 5 a d e^{3} + 5 b^{2} d f^{2} + 5 b^{2} e^{2} f + 5 b c^{2} f^{2} - 5 b c d e f - 5 b c e^{3} - 5 b d^{3} f + 5 b d^{2} e^{2} - 5 b f^{3} - 5 c^{3} e f + 5 c^{2} d^{2} f + 5 c^{2} d e^{2} - 5 c d^{3} e + 10 c e f^{2} + d^{5} + 5 d^{2} f^{2} - 15 d e^{2} f + 5 e^{4}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(E, -r0**5*r1**5*r2**5*r3**5*r4**5 - r0**5*r1**5*r2**5*r3**5*r5**5 - r0**5*r1**5*r2**5*r4**5*r5**5 - r0**5*r1**5*r3**5*r4**5*r5**5 - r0**5*r2**5*r3**5*r4**5*r5**5 - r1**5*r2**5*r3**5*r4**5*r5**5)"
],
"text/latex": [
"$\\displaystyle E = - r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} - r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{5}^{5} - r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{4}^{5} r_{5}^{5} - r_{0}^{5} r_{1}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5} - r_{0}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5} - r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(E, 5*a*f**4 - 5*b*e*f**3 - 5*c*d*f**3 + 5*c*e**2*f**2 + 5*d**2*e*f**2 - 5*d*e**3*f + e**5)"
],
"text/latex": [
"$\\displaystyle E = 5 a f^{4} - 5 b e f^{3} - 5 c d f^{3} + 5 c e^{2} f^{2} + 5 d^{2} e f^{2} - 5 d e^{3} f + e^{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(F, r0**5*r1**5*r2**5*r3**5*r4**5*r5**5)"
],
"text/latex": [
"$\\displaystyle F = r_{0}^{5} r_{1}^{5} r_{2}^{5} r_{3}^{5} r_{4}^{5} r_{5}^{5}$"
]
},
"metadata": {}
},
{
"output_type": "display_data",
"data": {
"text/plain": [
"Eq(F, f**5)"
],
"text/latex": [
"$\\displaystyle F = f^{5}$"
]
},
"metadata": {}
}
],
"metadata": {}
},
{
"cell_type": "code",
"execution_count": null,
"source": [],
"outputs": [],
"metadata": {}
}
],
"metadata": {
"orig_nbformat": 4,
"language_info": {
"name": "python",
"version": "3.9.5",
"mimetype": "text/x-python",
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"pygments_lexer": "ipython3",
"nbconvert_exporter": "python",
"file_extension": ".py"
},
"kernelspec": {
"name": "python3",
"display_name": "Python 3.9.5 64-bit"
},
"interpreter": {
"hash": "ac59ebe37160ed0dfa835113d9b8498d9f09ceb179beaac4002f036b9467c963"
}
},
"nbformat": 4,
"nbformat_minor": 2
}
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