files / problems / hilbert16-limit-cycles / compute / q2 / o-iterated-squaring / certs / preimages.json

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{
  "square_half_0": {
    "target": [
      1,
      2,
      0,
      1
    ],
    "real_regular": 2,
    "complex_affine": 4,
    "bezout_m2": 4,
    "attains_m2": false,
    "points": [
      {
        "u": "-sqrt(2)/2",
        "v": "0",
        "det": "2"
      },
      {
        "u": "sqrt(2)/2",
        "v": "0",
        "det": "2"
      }
    ]
  },
  "square_quarter_quarter": {
    "target": [
      1,
      4,
      1,
      4
    ],
    "real_regular": 2,
    "complex_affine": 4,
    "bezout_m2": 4,
    "attains_m2": false,
    "u2_quadratic": "64t^2 - 16t - 1",
    "disc": 512
  },
  "phi2_half_0": {
    "target": [
      1,
      2,
      0,
      1
    ],
    "real_regular": 4,
    "complexified_uv_at_most": 16,
    "holomorphic_in_C": 4,
    "note": "z^4 = 1/2 has 4 points in R^2 \u2245 C, not 16"
  },
  "chebyshev_T2_half_half": {
    "target": [
      1,
      2,
      1,
      2
    ],
    "real_regular": 4,
    "attains_m2": true,
    "note": "T2 x T2 is 4-to-1 on (-1,1)^2; components independent"
  },
  "polar_z_to_z_2k_of_half": [
    {
      "k": 1,
      "count": 2,
      "expected": 2
    },
    {
      "k": 2,
      "count": 4,
      "expected": 4
    },
    {
      "k": 3,
      "count": 8,
      "expected": 8
    },
    {
      "k": 4,
      "count": 16,
      "expected": 16
    },
    {
      "k": 5,
      "count": 32,
      "expected": 32
    },
    {
      "k": 6,
      "count": 64,
      "expected": 64
    }
  ],
  "honesty": {
    "real_plane_z_squared_is_2_to_1": true,
    "separable_T2_is_4_to_1_on_minus1_1_square": true,
    "CR_couples_the_components": true
  }
}