files / problems / hilbert16-limit-cycles / compute / q3 / pp-christopher-lloyd / certs / identity.json

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{
  "paper": "arXiv:2407.13465v2 \u00a74 / arXiv:2604.12883v1 Theorem 1 \u00a76",
  "Phi": "u^2, v^2",
  "det": "4uv",
  "time_rescale": "2uv",
  "field": "Yu=v P(u^2,v^2), Yv=u Q(u^2,v^2)",
  "not_adj": true,
  "adj_is_twice_cl": true,
  "identity_cl": "DPhi \u00b7 Y = 2uv (X o Phi)",
  "identity_adj": "DPhi \u00b7 Y_adj = 4uv (X o Phi)",
  "fprime_rho": [
    -1,
    2
  ],
  "linear": {
    "P": "y",
    "Q": "-x",
    "identity_ok": true,
    "deg_Y": 3,
    "bound": 3,
    "Yu_closed": "v^3",
    "Yv_closed": "-u^3"
  },
  "quadratic": {
    "P": "x^2+y",
    "Q": "y^2+x",
    "identity_ok": true,
    "deg_Y": 5,
    "bound": 5,
    "Yu_closed": "u^4 v + v^3",
    "Yv_closed": "u^3 + u v^4"
  },
  "radial": {
    "rho2": [
      1,
      4
    ],
    "shift": 2,
    "identity_ok": true,
    "deg_Y": 7,
    "bound": 7,
    "exact": true,
    "n_terms_Yu": 8,
    "n_terms_Yv": 8,
    "leading_Yu": "-u^6 v - u^2 v^5",
    "leading_Yv": "-u^5 v^2 - u v^6",
    "P_monomials": [
      {
        "u": 0,
        "v": 0,
        "num": 27,
        "den": 2
      },
      {
        "u": 0,
        "v": 1,
        "num": -7,
        "den": 1
      },
      {
        "u": 1,
        "v": 0,
        "num": -63,
        "den": 4
      },
      {
        "u": 0,
        "v": 2,
        "num": 2,
        "den": 1
      },
      {
        "u": 1,
        "v": 1,
        "num": 4,
        "den": 1
      },
      {
        "u": 2,
        "v": 0,
        "num": 6,
        "den": 1
      },
      {
        "u": 1,
        "v": 2,
        "num": -1,
        "den": 1
      },
      {
        "u": 3,
        "v": 0,
        "num": -1,
        "den": 1
      }
    ],
    "Q_monomials": [
      {
        "u": 0,
        "v": 0,
        "num": 35,
        "den": 2
      },
      {
        "u": 0,
        "v": 1,
        "num": -63,
        "den": 4
      },
      {
        "u": 1,
        "v": 0,
        "num": -9,
        "den": 1
      },
      {
        "u": 0,
        "v": 2,
        "num": 6,
        "den": 1
      },
      {
        "u": 1,
        "v": 1,
        "num": 4,
        "den": 1
      },
      {
        "u": 2,
        "v": 0,
        "num": 2,
        "den": 1
      },
      {
        "u": 0,
        "v": 3,
        "num": -1,
        "den": 1
      },
      {
        "u": 2,
        "v": 1,
        "num": -1,
        "den": 1
      }
    ],
    "Yu_monomials": [
      {
        "u": 0,
        "v": 1,
        "num": 27,
        "den": 2
      },
      {
        "u": 0,
        "v": 3,
        "num": -7,
        "den": 1
      },
      {
        "u": 2,
        "v": 1,
        "num": -63,
        "den": 4
      },
      {
        "u": 0,
        "v": 5,
        "num": 2,
        "den": 1
      },
      {
        "u": 2,
        "v": 3,
        "num": 4,
        "den": 1
      },
      {
        "u": 4,
        "v": 1,
        "num": 6,
        "den": 1
      },
      {
        "u": 2,
        "v": 5,
        "num": -1,
        "den": 1
      },
      {
        "u": 6,
        "v": 1,
        "num": -1,
        "den": 1
      }
    ],
    "Yv_monomials": [
      {
        "u": 1,
        "v": 0,
        "num": 35,
        "den": 2
      },
      {
        "u": 1,
        "v": 2,
        "num": -63,
        "den": 4
      },
      {
        "u": 3,
        "v": 0,
        "num": -9,
        "den": 1
      },
      {
        "u": 1,
        "v": 4,
        "num": 6,
        "den": 1
      },
      {
        "u": 3,
        "v": 2,
        "num": 4,
        "den": 1
      },
      {
        "u": 5,
        "v": 0,
        "num": 2,
        "den": 1
      },
      {
        "u": 1,
        "v": 6,
        "num": -1,
        "den": 1
      },
      {
        "u": 5,
        "v": 2,
        "num": -1,
        "den": 1
      }
    ]
  },
  "chebyshev_T2": {
    "T2": "2t^2-1",
    "deg_linear": 3,
    "deg_quad": 5,
    "deg_radial": 7,
    "n_terms_Yu": 8,
    "n_terms_Yv": 8,
    "Yu_monomials": [
      {
        "u": 0,
        "v": 1,
        "num": 3,
        "den": 1
      },
      {
        "u": 0,
        "v": 3,
        "num": -8,
        "den": 1
      },
      {
        "u": 2,
        "v": 1,
        "num": -30,
        "den": 1
      },
      {
        "u": 0,
        "v": 5,
        "num": 16,
        "den": 1
      },
      {
        "u": 2,
        "v": 3,
        "num": 32,
        "den": 1
      },
      {
        "u": 4,
        "v": 1,
        "num": 48,
        "den": 1
      },
      {
        "u": 2,
        "v": 5,
        "num": -32,
        "den": 1
      },
      {
        "u": 6,
        "v": 1,
        "num": -32,
        "den": 1
      }
    ],
    "Yv_monomials": [
      {
        "u": 1,
        "v": 0,
        "num": 11,
        "den": 1
      },
      {
        "u": 1,
        "v": 2,
        "num": -30,
        "den": 1
      },
      {
        "u": 3,
        "v": 0,
        "num": -24,
        "den": 1
      },
      {
        "u": 1,
        "v": 4,
        "num": 48,
        "den": 1
      },
      {
        "u": 3,
        "v": 2,
        "num": 32,
        "den": 1
      },
      {
        "u": 5,
        "v": 0,
        "num": 16,
        "den": 1
      },
      {
        "u": 1,
        "v": 6,
        "num": -32,
        "den": 1
      },
      {
        "u": 5,
        "v": 2,
        "num": -32,
        "den": 1
      }
    ]
  }
}