files / problems / hilbert16-limit-cycles / compute / q3 / jj-weak-hilbert / certs / family.json

2.9 KB · on GitHub

{
  "claim": "Radial family on H=(x^2+y^2)/2 attains Z(2,n)=floor((n-1)/2) and does not beat it. Not a bound on H(2).",
  "formula_beaten": false,
  "hamiltonian": {
    "Hnum": [
      {
        "coeff": "1",
        "y": 2
      },
      {
        "coeff": "1",
        "x": 2
      }
    ],
    "P": [
      {
        "coeff": "1",
        "y": 1
      }
    ],
    "Qfield": [
      {
        "coeff": "-1",
        "x": 1
      }
    ],
    "dHdt": [],
    "variables": [
      "x",
      "y"
    ]
  },
  "hn_moved": false,
  "n3": {
    "I_tilde": [
      0,
      1,
      -2
    ],
    "Q": [
      {
        "coeff": "1",
        "y": 1
      },
      {
        "coeff": "-1",
        "y": 3
      },
      {
        "coeff": "-1",
        "x": 2,
        "y": 1
      }
    ],
    "Z": 1,
    "degQ": 3,
    "p": [
      1,
      -1
    ],
    "positive_zeros_h": [
      "1/2"
    ],
    "variables": [
      "x",
      "y"
    ]
  },
  "n5": {
    "I_tilde": [
      0,
      4,
      -10,
      4
    ],
    "Q": [
      {
        "coeff": "4",
        "y": 1
      },
      {
        "coeff": "-5",
        "y": 3
      },
      {
        "coeff": "1",
        "y": 5
      },
      {
        "coeff": "-5",
        "x": 2,
        "y": 1
      },
      {
        "coeff": "2",
        "x": 2,
        "y": 3
      },
      {
        "coeff": "1",
        "x": 4,
        "y": 1
      }
    ],
    "Z": 2,
    "degQ": 5,
    "p": [
      4,
      -5,
      1
    ],
    "positive_zeros_h": [
      "1/2",
      "2"
    ],
    "variables": [
      "x",
      "y"
    ]
  },
  "not_H2": true,
  "record": "Scholarpedia Han-Li-Li 2010; Z(2,n)=floor((n-1)/2) because M(h) is a polynomial",
  "rows": [
    {
      "Z": 0,
      "beat_degQ": 3,
      "degQ": 1,
      "matches": 1,
      "n": 1,
      "zeros": 0
    },
    {
      "Z": 0,
      "beat_degQ": 3,
      "degQ": 1,
      "matches": 1,
      "n": 2,
      "zeros": 0
    },
    {
      "Z": 1,
      "beat_degQ": 5,
      "degQ": 3,
      "matches": 1,
      "n": 3,
      "zeros": 1
    },
    {
      "Z": 1,
      "beat_degQ": 5,
      "degQ": 3,
      "matches": 1,
      "n": 4,
      "zeros": 1
    },
    {
      "Z": 2,
      "beat_degQ": 7,
      "degQ": 5,
      "matches": 1,
      "n": 5,
      "zeros": 2
    },
    {
      "Z": 2,
      "beat_degQ": 7,
      "degQ": 5,
      "matches": 1,
      "n": 6,
      "zeros": 2
    },
    {
      "Z": 3,
      "beat_degQ": 9,
      "degQ": 7,
      "matches": 1,
      "n": 7,
      "zeros": 3
    },
    {
      "Z": 3,
      "beat_degQ": 9,
      "degQ": 7,
      "matches": 1,
      "n": 8,
      "zeros": 3
    },
    {
      "Z": 4,
      "beat_degQ": 11,
      "degQ": 9,
      "matches": 1,
      "n": 9,
      "zeros": 4
    },
    {
      "Z": 4,
      "beat_degQ": 11,
      "degQ": 9,
      "matches": 1,
      "n": 10,
      "zeros": 4
    }
  ],
  "schema": "hilbert16-jj-weak-hilbert/v1"
}