files / problems / hilbert16-limit-cycles / compute / q3 / ii-complex-cube / certs / preimages.json

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{
  "cube_half_0": {
    "target": [
      1,
      2,
      0,
      1
    ],
    "real_regular": 3,
    "complex_affine": 9,
    "bezout_m2": 9,
    "attains_m2": false,
    "points": [
      {
        "u": "-2**(2/3)/4",
        "v": "-2**(2/3)*sqrt(3)/4",
        "det": "9*2**(2/3)/4"
      },
      {
        "u": "-2**(2/3)/4",
        "v": "2**(2/3)*sqrt(3)/4",
        "det": "9*2**(2/3)/4"
      },
      {
        "u": "2**(2/3)/2",
        "v": "0",
        "det": "9*2**(2/3)/4"
      }
    ]
  },
  "cube_quarter_quarter": {
    "target": [
      1,
      4,
      1,
      4
    ],
    "real_regular": 3,
    "complex_affine": 9,
    "bezout_m2": 9,
    "attains_m2": false,
    "seed_point": [
      -1,
      2,
      1,
      2
    ]
  },
  "chebyshev_T3_half_half": {
    "target": [
      1,
      2,
      1,
      2
    ],
    "real_regular": 9,
    "attains_m2": true,
    "t3_cubic": "8t^3-6t-1",
    "disc": 5184,
    "note": "T3 x T3 is 9-to-1 on (-1,1)^2; components independent"
  },
  "polar_z_to_z3": [
    {
      "target": [
        1,
        2,
        0,
        1
      ],
      "count": 3
    },
    {
      "target": [
        1,
        4,
        1,
        4
      ],
      "count": 3
    }
  ],
  "resultant": {
    "Res_v": "64u^9-48x u^6-15x^2 u^3-27y^2 u^3-x^3",
    "deg_u": 9,
    "deg_t": 3,
    "cubic_in_u3": true,
    "bezout_complex": 9
  },
  "honesty": {
    "real_plane_z_cubed_is_3_to_1": true,
    "separable_T3_is_9_to_1_on_minus1_1_square": true,
    "CR_couples_the_components": true
  }
}