The conjectural presentation of the valuation algebra of complex space forms

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Define the formal series \fk(s,t,λ)\f_k(s,t,\lambda) by

log⁡(1+sx2+tx+∑n[(4n+1n+1)−9(4n+1n−1)]λnx−2n)=∑kfˉk(s,t,λ)xk.\log\left(1+s x^2+t x+\sum_{n}\left[\binom{4n+1}{n+1}-9\binom{4n+1}{n-1}\right]\lambda^n x^{-2n}\right)=\sum_k \bar f_k(s,t,\lambda)x^k.

The presentation conjecture. For the valuation algebra Vλn\mathcal{V}^n_\lambda,

Vλn≃R[s,t]/(fˉn+1,fˉn+2,t2n+1,st2n−1,…,snt).\mathcal{V}^n_\lambda\simeq \mathbb{R}[s,t]/(\bar f_{n+1},\bar f_{n+2},t^{2n+1},st^{2n-1},\dots,s^nt).

This was the first conjectural description of the algebras Vλn\mathcal{V}^n_\lambda for λ≠0\lambda\ne0, before the isomorphism and structure theorems established that the algebras for fixed nn and varying λ\lambda are isomorphic as filtered algebras. The source gives no explicit resolution status for this initial conjectural presentation.

References

Primary source

Andreas Bernig, Joseph H. G. Fu and Gil Solanes, “Integral geometry of complex space forms”, arXiv:1204.0604 (2013).

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