files / problems / hilbert16-limit-cycles / compute / q3 / hh-qh-melnikov / certs / identities.json

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{
  "beats_H3": false,
  "claim": "first-order Melnikov of the named cubic family P=0, Q=mu (alpha-x^2) y on the qh center has at most one positive zero; not a bound on H(3)",
  "exact": {
    "dx": [],
    "variables": [
      "x"
    ],
    "x2dx_cleared": []
  },
  "family": {
    "P": [],
    "Q": [
      {
        "alpha": 1,
        "coeff": "1",
        "mu": 1,
        "y": 1
      },
      {
        "coeff": "-1",
        "mu": 1,
        "x": 2,
        "y": 1
      }
    ],
    "Q_mu_y_extra": [
      {
        "coeff": "8",
        "mu": 1,
        "y": 2
      }
    ],
    "claimed": [
      {
        "alpha": 1,
        "coeff": "8",
        "mu": 1,
        "y": 2
      },
      {
        "coeff": "-8",
        "mu": 1,
        "x": 2,
        "y": 2
      }
    ],
    "div": [
      {
        "alpha": 1,
        "coeff": "1",
        "mu": 1
      },
      {
        "coeff": "-1",
        "mu": 1,
        "x": 2
      }
    ],
    "extra": [
      {
        "alpha": 1,
        "coeff": "8",
        "mu": 1,
        "y": 2
      },
      {
        "coeff": "-8",
        "mu": 1,
        "x": 2,
        "y": 2
      }
    ],
    "variables": [
      "x",
      "y",
      "mu",
      "alpha"
    ]
  },
  "fourteen_zeros": false,
  "general_cubic_I_zeros_at_most": 2,
  "hn_moved": false,
  "melnikov": {
    "divergence": "-iint (P_x + Q_y) dA",
    "energy_factor_when_P_eq_0": 4,
    "formula": "oint Q dx - P dy"
  },
  "named_family_cyclicity_at_most": 1,
  "rectangles": {
    "area_den": 16,
    "area_num": 5,
    "moment_den": 256,
    "moment_num": 97
  },
  "scale": {
    "H_diff": [],
    "J0_exponent": 3,
    "J2_exponent": 5,
    "Jy2_exponent": 7,
    "integrand_1": [],
    "integrand_x2": [],
    "integrand_y2": [],
    "jac_diff": [],
    "ratio_diff": [],
    "ratio_exponent": 2,
    "variables": [
      "s",
      "u",
      "lam"
    ]
  },
  "schema": "hilbert16-hh-qh-melnikov/v1",
  "unperturbed": {
    "H": [
      {
        "coeff": "4",
        "y": 2
      },
      {
        "coeff": "1",
        "x": 4
      }
    ],
    "Hx": [
      {
        "coeff": "4",
        "x": 3
      }
    ],
    "Hy": [
      {
        "coeff": "8",
        "y": 1
      }
    ],
    "P0": [
      {
        "coeff": "2",
        "y": 1
      }
    ],
    "Q0": [
      {
        "coeff": "-1",
        "x": 3
      }
    ],
    "dHdt": [],
    "hess11": [
      {
        "coeff": "12",
        "x": 2
      }
    ],
    "hess12": [],
    "hess22": [
      {
        "coeff": "8"
      }
    ],
    "j11": [],
    "j12": [
      {
        "coeff": "2"
      }
    ],
    "j21": [
      {
        "coeff": "-3",
        "x": 2
      }
    ],
    "j22": [],
    "origin_det": 0,
    "origin_hess_det": 0,
    "origin_trace": 0,
    "sos": [
      {
        "coeff": "1",
        "y": 2
      },
      {
        "coeff": "1",
        "x": 6
      }
    ],
    "variables": [
      "x",
      "y"
    ]
  },
  "weight": {
    "H_weight_diff": [],
    "P_weight_diff": [],
    "Q_weight_diff": [],
    "variables": [
      "x",
      "y",
      "lam"
    ]
  },
  "what_this_is_not": [
    "not a dent of H(3)",
    "not fourteen zeros of I(h)",
    "not a beat of Gavrilov-He-Xiao arXiv:2606.22137"
  ]
}