Tropical Nevanlinna theory of several variables

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 Rⁿ into tropical projective space TPᵐ intersecting tropical hypersurfaces ⱼ\ⱼ₌₁q with degree dⱼ, we then obtain the Second Main Theorem \|\,\,\, (q-M-1-λ)Tf(r) ≤ ∑j=M+2q 1/dⱼN(r,1T ⊘ Pⱼ ∘ f) + o(Tf(r)), where d=lcd(d₁, …, dq) and M=(dᵐ⁺ᵈ)-1.