files / problems / hilbert16-limit-cycles / compute / q2 / f-homogeneous / certs / identities.json

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{
  "circles": {
    "F": [],
    "G": [
      {
        "coeff": "1",
        "y": 4
      },
      {
        "coeff": "2",
        "x": 2,
        "y": 2
      },
      {
        "coeff": "1",
        "x": 4
      }
    ],
    "G_claimed": [
      {
        "coeff": "1",
        "y": 4
      },
      {
        "coeff": "2",
        "x": 2,
        "y": 2
      },
      {
        "coeff": "1",
        "x": 4
      }
    ],
    "P": [
      {
        "coeff": "-1",
        "y": 3
      },
      {
        "coeff": "-1",
        "x": 2,
        "y": 1
      }
    ],
    "Q": [
      {
        "coeff": "1",
        "x": 1,
        "y": 2
      },
      {
        "coeff": "1",
        "x": 3
      }
    ],
    "variables": [
      "x",
      "y"
    ]
  },
  "claim": "unperturbed homogeneous and quasi-homogeneous planar fields have zero isolated periodic orbits; not a bound on H(n)",
  "homog2": {
    "F": [
      {
        "b02": 1,
        "coeff": "1",
        "y": 3
      },
      {
        "b11": 1,
        "coeff": "1",
        "x": 1,
        "y": 2
      },
      {
        "a02": 1,
        "coeff": "1",
        "x": 1,
        "y": 2
      },
      {
        "b20": 1,
        "coeff": "1",
        "x": 2,
        "y": 1
      },
      {
        "a11": 1,
        "coeff": "1",
        "x": 2,
        "y": 1
      },
      {
        "a20": 1,
        "coeff": "1",
        "x": 3
      }
    ],
    "F_scaled_diff": [],
    "G": [
      {
        "a02": 1,
        "coeff": "-1",
        "y": 3
      },
      {
        "b02": 1,
        "coeff": "1",
        "x": 1,
        "y": 2
      },
      {
        "a11": 1,
        "coeff": "-1",
        "x": 1,
        "y": 2
      },
      {
        "b11": 1,
        "coeff": "1",
        "x": 2,
        "y": 1
      },
      {
        "a20": 1,
        "coeff": "-1",
        "x": 2,
        "y": 1
      },
      {
        "b20": 1,
        "coeff": "1",
        "x": 3
      }
    ],
    "G_scaled_diff": [],
    "P": [
      {
        "a02": 1,
        "coeff": "1",
        "y": 2
      },
      {
        "a11": 1,
        "coeff": "1",
        "x": 1,
        "y": 1
      },
      {
        "a20": 1,
        "coeff": "1",
        "x": 2
      }
    ],
    "P_scaled_diff": [],
    "Q": [
      {
        "b02": 1,
        "coeff": "1",
        "y": 2
      },
      {
        "b11": 1,
        "coeff": "1",
        "x": 1,
        "y": 1
      },
      {
        "b20": 1,
        "coeff": "1",
        "x": 2
      }
    ],
    "Q_scaled_diff": [],
    "variables": [
      "x",
      "y",
      "lam",
      "a20",
      "a11",
      "a02",
      "b20",
      "b11",
      "b02"
    ]
  },
  "homog3": {
    "F": [
      {
        "b03": 1,
        "coeff": "1",
        "y": 4
      },
      {
        "b12": 1,
        "coeff": "1",
        "x": 1,
        "y": 3
      },
      {
        "a03": 1,
        "coeff": "1",
        "x": 1,
        "y": 3
      },
      {
        "b21": 1,
        "coeff": "1",
        "x": 2,
        "y": 2
      },
      {
        "a12": 1,
        "coeff": "1",
        "x": 2,
        "y": 2
      },
      {
        "b30": 1,
        "coeff": "1",
        "x": 3,
        "y": 1
      },
      {
        "a21": 1,
        "coeff": "1",
        "x": 3,
        "y": 1
      },
      {
        "a30": 1,
        "coeff": "1",
        "x": 4
      }
    ],
    "F_scaled_diff": [],
    "G": [
      {
        "a03": 1,
        "coeff": "-1",
        "y": 4
      },
      {
        "b03": 1,
        "coeff": "1",
        "x": 1,
        "y": 3
      },
      {
        "a12": 1,
        "coeff": "-1",
        "x": 1,
        "y": 3
      },
      {
        "b12": 1,
        "coeff": "1",
        "x": 2,
        "y": 2
      },
      {
        "a21": 1,
        "coeff": "-1",
        "x": 2,
        "y": 2
      },
      {
        "b21": 1,
        "coeff": "1",
        "x": 3,
        "y": 1
      },
      {
        "a30": 1,
        "coeff": "-1",
        "x": 3,
        "y": 1
      },
      {
        "b30": 1,
        "coeff": "1",
        "x": 4
      }
    ],
    "G_scaled_diff": [],
    "P": [
      {
        "a03": 1,
        "coeff": "1",
        "y": 3
      },
      {
        "a12": 1,
        "coeff": "1",
        "x": 1,
        "y": 2
      },
      {
        "a21": 1,
        "coeff": "1",
        "x": 2,
        "y": 1
      },
      {
        "a30": 1,
        "coeff": "1",
        "x": 3
      }
    ],
    "P_scaled_diff": [],
    "Q": [
      {
        "b03": 1,
        "coeff": "1",
        "y": 3
      },
      {
        "b12": 1,
        "coeff": "1",
        "x": 1,
        "y": 2
      },
      {
        "b21": 1,
        "coeff": "1",
        "x": 2,
        "y": 1
      },
      {
        "b30": 1,
        "coeff": "1",
        "x": 3
      }
    ],
    "Q_scaled_diff": [],
    "variables": [
      "x",
      "y",
      "lam",
      "a30",
      "a21",
      "a12",
      "a03",
      "b30",
      "b21",
      "b12",
      "b03"
    ]
  },
  "quasihomogeneous": {
    "H": [
      {
        "coeff": "4",
        "y": 2
      },
      {
        "coeff": "1",
        "x": 4
      }
    ],
    "Hx": [
      {
        "coeff": "4",
        "x": 3
      }
    ],
    "Hy": [
      {
        "coeff": "8",
        "y": 1
      }
    ],
    "P": [
      {
        "coeff": "2",
        "y": 1
      }
    ],
    "Q": [
      {
        "coeff": "-1",
        "x": 3
      }
    ],
    "dHdt": [],
    "variables": [
      "x",
      "y"
    ]
  },
  "quasihomogeneous_weight": {
    "H_weight_diff": [],
    "P_weight_diff": [],
    "Q_weight_diff": [],
    "variables": [
      "x",
      "y",
      "lam"
    ]
  },
  "rays": {
    "F": [
      {
        "coeff": "1",
        "y": 3
      },
      {
        "coeff": "1",
        "x": 3
      }
    ],
    "F_zero_line": [],
    "F_zero_line_normal": [
      {
        "coeff": "2",
        "x": 2
      }
    ],
    "G": [
      {
        "coeff": "1",
        "x": 1,
        "y": 2
      },
      {
        "coeff": "-1",
        "x": 2,
        "y": 1
      }
    ],
    "G_factored": [
      {
        "coeff": "1",
        "x": 1,
        "y": 2
      },
      {
        "coeff": "-1",
        "x": 2,
        "y": 1
      }
    ],
    "P": [
      {
        "coeff": "1",
        "x": 2
      }
    ],
    "Q": [
      {
        "coeff": "1",
        "y": 2
      }
    ],
    "ray_x0": [],
    "ray_y0": [],
    "ray_yx": [],
    "variables": [
      "x",
      "y"
    ]
  },
  "scale": {
    "P": [
      {
        "coeff": "-1",
        "y": 3
      }
    ],
    "P_scaled_diff": [],
    "Q": [
      {
        "coeff": "1",
        "x": 3
      }
    ],
    "Q_scaled_diff": [],
    "variables": [
      "x",
      "y",
      "lam"
    ]
  },
  "schema": "hilbert16-f-homogeneous/v1"
}