A Counterexample to the Jacobian Conjecture

yacobianconj.ipynb

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      "det J = -2\n",
      "expanded:\n",
      "f = x**3*y**3*z + 3*x**2*y**4 + 3*x**2*y**2*z + 7*x*y**3 + 3*x*y*z + 4*y**2 + z\n",
      "g = 3*x**3*y**2*z + 9*x**2*y**3 + 6*x**2*y*z + 12*x*y**2 + 3*x*z + y\n",
      "h = -x**3*z - 3*x**2*y + 2*x\n",
      "Jacobian:\n",
      "⎡         3              3                              2             2        ↪\n",
      "⎢      3⋅y ⋅(x⋅y + 1) + y ⋅(3⋅x⋅y + 4) + 3⋅y⋅z⋅(x⋅y + 1)         3⋅x⋅y ⋅(x⋅y + ↪\n",
      "⎢                                                                              ↪\n",
      "⎢     3                          2                            2                ↪\n",
      "⎢9⋅x⋅y  + 6⋅x⋅y⋅z⋅(x⋅y + 1) + 3⋅y ⋅(3⋅x⋅y + 4) + 3⋅z⋅(x⋅y + 1)                 ↪\n",
      "⎢                                                                              ↪\n",
      "⎢                          2                                                   ↪\n",
      "⎣                     - 3⋅x ⋅z - 6⋅x⋅y + 2                                     ↪\n",
      "\n",
      "↪          2                              2                                    ↪\n",
      "↪  1) + x⋅y ⋅(3⋅x⋅y + 4) + 3⋅x⋅z⋅(x⋅y + 1)  + 2⋅y⋅(x⋅y + 1)⋅(3⋅x⋅y + 4)    (x⋅ ↪\n",
      "↪                                                                              ↪\n",
      "↪       2  2      2                                                            ↪\n",
      "↪    9⋅x ⋅y  + 6⋅x ⋅z⋅(x⋅y + 1) + 6⋅x⋅y⋅(3⋅x⋅y + 4) + 1                  3⋅x⋅( ↪\n",
      "↪                                                                              ↪\n",
      "↪                              2                                               ↪\n",
      "↪                          -3⋅x                                                ↪\n",
      "\n",
      "↪       3  ⎤\n",
      "↪ y + 1)   ⎥\n",
      "↪          ⎥\n",
      "↪         2⎥\n",
      "↪ x⋅y + 1) ⎥\n",
      "↪          ⎥\n",
      "↪   3      ⎥\n",
      "↪ -x       ⎦\n",
      "(0, 0, -1/4) -> (-1/4, 0, 0)\n",
      "(1, -3/2, 13/2) -> (-1/4, 0, 0)\n",
      "(-1, 3/2, 13/2) -> (-1/4, 0, 0)\n"
     ]
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   "source": [
    "import sympy as sp\n",
    "\n",
    "x, y, z = sp.symbols('x y z')\n",
    "\n",
    "f = (1+x*y)**3*z + y**2*(1+x*y)*(4+3*x*y)\n",
    "g = y + 3*x*(1+x*y)**2*z + 3*x*y**2*(4+3*x*y)\n",
    "h = 2*x - 3*x**2*y - x**3*z\n",
    "\n",
    "F = sp.Matrix([f, g, h])\n",
    "J = F.jacobian([x,y,z])\n",
    "detJ = sp.factor(J.det())\n",
    "\n",
    "fe, ge, he = map(sp.expand, (f, g, h))\n",
    "\n",
    "print(\"det J =\", detJ)\n",
    "print(\"expanded:\")\n",
    "print(\"f =\", fe)\n",
    "print(\"g =\", ge)\n",
    "print(\"h =\", he)\n",
    "print(\"Jacobian:\")\n",
    "\n",
    "sp.pprint(J)\n",
    "\n",
    "points = [\n",
    "    (0, 0, sp.Rational(-1, 4)),\n",
    "    (1, sp.Rational(-3, 2), sp.Rational(13, 2)),\n",
    "    (-1, sp.Rational(3, 2), sp.Rational(13, 2)),\n",
    "]\n",
    "\n",
    "for p in points:\n",
    "    image = F.subs(dict(zip((x, y, z), p))).applyfunc(sp.simplify)\n",
    "    print(p, \"->\", tuple(image))"
   ]
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