Tropical Griffiths-type defect relation for tropical hypersurfaces

From papers

Let qq and nn be positive integers with qnq\geq n. Let f:RTPnf:\mathbb{R}\rightarrow\mathbb{TP}^{n} be a tropical algebraically nondegenerated tropical holomorphic curve. Assume that PjP_j (j=1,,q)(j=1,\ldots,q) are homogeneous tropical polynomials of degrees djd_j, and let dd be the least common multiple of d1,,dqd_1,\ldots,d_q. Set M=(n+dd)1M=\binom{n+d}{d}-1. If λ=ddg({PM+2f,,Pqf})\lambda=\operatorname{ddg}(\{P_{M+2}\circ f,\ldots,P_q\circ f\}) and

lim suprlogTf(r)r=0,\limsup_{r\rightarrow\infty}\frac{\log T_f(r)}{r}=0,

Tropical Griffiths-type defect conjecture. One should have

j=1qδf(VPj)n+1+λd.\sum_{j=1}^{q}\delta_f(V_{P_j})\leq\frac{n+1+\lambda}{d}.

This is proposed as a tropical analogue of the Griffiths conjecture on defect relations for holomorphic curves and hypersurfaces. The source notes that related classical results are known in special cases, but does not state a resolution of this tropical formulation.

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Sources & referencesView supporting material

Primary source

Tingbin Cao and Jianhua Zheng, “Second main theorem with tropical hypersurfaces and defect relation”, arXiv:1806.04294 (2018).

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