files / problems / hilbert16-limit-cycles / compute / q3 / ii-complex-cube / certs / identity.json

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{
  "paper": "arXiv:2604.12883v1 Remark 4 / Theorem 1 / Theorem 2 / \u00a76",
  "Phi": "u^3-3uv^2, 3u^2v-v^3",
  "det": "9(u^2+v^2)^2",
  "cauchy_riemann": true,
  "modulus_identity": true,
  "identity": "DPhi \u00b7 Y = (det DPhi) (X o Phi)",
  "linear": {
    "P": "y",
    "Q": "-x",
    "identity_ok": true,
    "deg_Y": 5,
    "bound": 5,
    "Yu_closed": "3v(u^2+v^2)^2",
    "Yv_closed": "-3u(u^2+v^2)^2",
    "Yu_monomials": [
      {
        "u": 0,
        "v": 5,
        "num": 3,
        "den": 1
      },
      {
        "u": 2,
        "v": 3,
        "num": 6,
        "den": 1
      },
      {
        "u": 4,
        "v": 1,
        "num": 3,
        "den": 1
      }
    ],
    "Yv_monomials": [
      {
        "u": 1,
        "v": 4,
        "num": -3,
        "den": 1
      },
      {
        "u": 3,
        "v": 2,
        "num": -6,
        "den": 1
      },
      {
        "u": 5,
        "v": 0,
        "num": -3,
        "den": 1
      }
    ]
  },
  "quadratic": {
    "P": "x^2+y",
    "Q": "y^2+x",
    "identity_ok": true,
    "deg_Y": 8,
    "bound": 8,
    "n_terms_Yu": 10,
    "n_terms_Yv": 10
  },
  "radial": {
    "rho2": [
      1,
      4
    ],
    "identity_ok": true,
    "deg_Y": 11,
    "bound": 11,
    "exact": true,
    "n_terms_Yu": 12,
    "n_terms_Yv": 12,
    "leading_Yu": "-3u(u^2+v^2)^5",
    "leading_Yv": "-3v(u^2+v^2)^5",
    "Yu_monomials": [
      {
        "u": 0,
        "v": 5,
        "num": 3,
        "den": 1
      },
      {
        "u": 1,
        "v": 4,
        "num": 3,
        "den": 4
      },
      {
        "u": 2,
        "v": 3,
        "num": 6,
        "den": 1
      },
      {
        "u": 3,
        "v": 2,
        "num": 3,
        "den": 2
      },
      {
        "u": 4,
        "v": 1,
        "num": 3,
        "den": 1
      },
      {
        "u": 5,
        "v": 0,
        "num": 3,
        "den": 4
      },
      {
        "u": 1,
        "v": 10,
        "num": -3,
        "den": 1
      },
      {
        "u": 3,
        "v": 8,
        "num": -15,
        "den": 1
      },
      {
        "u": 5,
        "v": 6,
        "num": -30,
        "den": 1
      },
      {
        "u": 7,
        "v": 4,
        "num": -30,
        "den": 1
      },
      {
        "u": 9,
        "v": 2,
        "num": -15,
        "den": 1
      },
      {
        "u": 11,
        "v": 0,
        "num": -3,
        "den": 1
      }
    ],
    "Yv_monomials": [
      {
        "u": 0,
        "v": 5,
        "num": 3,
        "den": 4
      },
      {
        "u": 1,
        "v": 4,
        "num": -3,
        "den": 1
      },
      {
        "u": 2,
        "v": 3,
        "num": 3,
        "den": 2
      },
      {
        "u": 3,
        "v": 2,
        "num": -6,
        "den": 1
      },
      {
        "u": 4,
        "v": 1,
        "num": 3,
        "den": 4
      },
      {
        "u": 5,
        "v": 0,
        "num": -3,
        "den": 1
      },
      {
        "u": 0,
        "v": 11,
        "num": -3,
        "den": 1
      },
      {
        "u": 2,
        "v": 9,
        "num": -15,
        "den": 1
      },
      {
        "u": 4,
        "v": 7,
        "num": -30,
        "den": 1
      },
      {
        "u": 6,
        "v": 5,
        "num": -30,
        "den": 1
      },
      {
        "u": 8,
        "v": 3,
        "num": -15,
        "den": 1
      },
      {
        "u": 10,
        "v": 1,
        "num": -3,
        "den": 1
      }
    ]
  },
  "chebyshev_T3": {
    "deg_linear": 5,
    "deg_quad": 8,
    "deg_radial": 11,
    "identity_ok": true
  }
}