Tropical Nevanlinna theory of several variables

Abstract

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 $$\Vert (q-M-1-\lambda )T_f(r)\le \sum _{j=M+2}^q\frac{1}{d_j}N(r,1_{\mathbb{T}}\oslash P_j\circ f)+o(T_f(r)),$$ where $d=lcd(d_1,\dots ,d_q)$ and $M={(}_d^{m+d})-1.$

46 pages

References in corpus (1)

• Tropical Nevanlinna theory and second main theorem