files / problems / hilbert16-limit-cycles / compute / q3 / ll-invariant-line / certs / core.json

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{
  "P_axis_at_1": 17,
  "bendixson_inconclusive": true,
  "certified_cycle_count": 0,
  "claim": "Named cubic dx/dt=16y+16x+x^3, dy/dt=16xy has invariant line y=0 (not a line of equilibria) and, by Dulac B=1/y, exactly 0 isolated periodic orbits. The same count holds for the family dx/dt=y+x+mu x^3, dy/dt=xy with mu>=0. Three cycles were not produced. Ye/Cherkas uniqueness for quadratics is context, not re-proved. Not a bound on H(n).",
  "cofactor_family": "x",
  "cofactor_named": "16x",
  "degeneration_is_axis_of_equilibria": true,
  "degree": 3,
  "det_origin": 0,
  "div_at_0": 16,
  "div_at_minus3": -5,
  "div_disc": 64,
  "dulac": "1/y",
  "dulac_num_min": 16,
  "dulac_numerator_family": "1+3 mu x^2",
  "dulac_numerator_named": "16+3x^2",
  "equilibria_count": 1,
  "family": "dx/dt = y+x+mu x^3, dy/dt = xy (mu>=0)",
  "family_cycle_count": 0,
  "family_degree_mu0": 2,
  "field": "dx/dt = 16y+16x+x^3, dy/dt = 16xy",
  "hn_moved": false,
  "invariant_line": "y=0",
  "line_of_equilibria": false,
  "named_mu_disc": "1/4",
  "named_mu_disc_den": 16,
  "named_mu_disc_num": 4,
  "schema": "hilbert16-ll-invariant-line-core/v1",
  "three_cycles_produced": false,
  "trace_origin": 16,
  "what_this_is_not": [
    "not a dent of H(n)",
    "not three cubic cycles",
    "not a re-proof of Ye/Cherkas",
    "not a line of equilibria",
    "not a bound on every cubic with a line"
  ],
  "ye_quadratic_at_most_one": true,
  "ye_reproved": false
}