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| Format: | Preprint |
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2025
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| Accès en ligne: | https://arxiv.org/abs/2508.20480 |
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| _version_ | 1866915467839406080 |
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| author | Cao, Tingbin Peng, Jiahu |
| author_facet | Cao, Tingbin Peng, Jiahu |
| contents | The main goal of this paper is to establish the higher-dimensional Nevanlinna theory in tropical geometry. We first develop a theory of tropical meromorphic functions ( holomorphic maps) in several variables, such as the proximity function, counting function and characteristic function, the first main theorem, higher-dimensional tropical versions of the logarithmic derivative lemmas. Based on this, for algebraically nondegenerate tropical holomorphic maps $f$ with subnormal growth from $\mathbb{R}^n$ into tropical projective space $\mathbb{TP}^{m}$ intersecting tropical hypersurfaces $\{V_{P_j}\}_{j=1}^{q}$ with degree $d_{j},$ we then obtain the Second Main Theorem $$\|\,\,\, (q-M-1-λ)T_f(r) \leq \sum_{j=M+2}^q \tfrac{1}{d_j}N(r,1_{\mathbb{T}} \oslash P_j \circ f) + o(T_f(r)),$$ where $d=lcd(d_{1}, \ldots, d_{q})$ and $M=(_d^{m+d})-1.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_20480 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Tropical Nevanlinna theory of several variables Cao, Tingbin Peng, Jiahu Algebraic Geometry 14T10, 32H30 The main goal of this paper is to establish the higher-dimensional Nevanlinna theory in tropical geometry. We first develop a theory of tropical meromorphic functions ( holomorphic maps) in several variables, such as the proximity function, counting function and characteristic function, the first main theorem, higher-dimensional tropical versions of the logarithmic derivative lemmas. Based on this, for algebraically nondegenerate tropical holomorphic maps $f$ with subnormal growth from $\mathbb{R}^n$ into tropical projective space $\mathbb{TP}^{m}$ intersecting tropical hypersurfaces $\{V_{P_j}\}_{j=1}^{q}$ with degree $d_{j},$ we then obtain the Second Main Theorem $$\|\,\,\, (q-M-1-λ)T_f(r) \leq \sum_{j=M+2}^q \tfrac{1}{d_j}N(r,1_{\mathbb{T}} \oslash P_j \circ f) + o(T_f(r)),$$ where $d=lcd(d_{1}, \ldots, d_{q})$ and $M=(_d^{m+d})-1.$ |
| title | Tropical Nevanlinna theory of several variables |
| topic | Algebraic Geometry 14T10, 32H30 |
| url | https://arxiv.org/abs/2508.20480 |