Truncated tropical Cartan second main theorem for 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 nondegerated tropical holomorphic curve. Assume that VPjV_{P_j} are defined by homogeneous tropical polynomials PjP_j (j=1,,q)(j=1,\ldots,q) of degree 11, respectively. If λ=ddg({Pn+2f,,Pqf})\lambda=\operatorname{ddg}(\{P_{n+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,

then, outside an exceptional set of zero upper density measure,

(qn1λ)Tf(r)j=n+2qNn)(r,1oPjf)+o(Tf(r)).(q-n-1-\lambda)T_f(r)\leq\sum_{j=n+2}^{q}N^{n)}\left(r,\frac{1_o}{P_j\circ f}\oslash\right)+o(T_f(r)).

In the special case λ=0\lambda=0, one should have, outside an exceptional set of zero upper density measure,

(qn1)Tf(r)=j=n+2qNn)(r,1oPjf)+o(Tf(r)).(q-n-1)T_f(r)=\sum_{j=n+2}^{q}N^{n)}\left(r,\frac{1_o}{P_j\circ f}\oslash\right)+o(T_f(r)).

Truncated tropical Cartan conjecture. The displayed truncated second main theorem is proposed as the tropical counterpart of the truncated Cartan second main theorem. The source gives no proof or resolution of this 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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