Tropical variants of some complex analysis results

Li, Kai; Laine, Ilpo; Tohge, Kazuya

doi:10.5186/aasfm.2016.4159

Cited by 2 publications

(1 citation statement)

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“…After modifying the Halburd-Southall's version of Tropical Poisson-Jensen theore and first main theorem, Laine and Tohge gave the tropical Nevanlinna second main theorem for tropical meromorphic functions with hyperorder strictly less than one. Laine, Liu and Tohge [16] presented tropical counterparts of some classical complex results related to Fermat type equations, Hayman conjecture and Brück conjecture. Recently, Korhonen and Tohge [15] extended the tropical Nevanlinna theory to tropical holomorphic curves in a finite dimensional tropical projective space, and obtained a tropical version of Cartan second main theorem for tropical holomorphic curves with hyperorder strictly less than one.…”

Section: Introductionmentioning

confidence: 99%

Second main theorem with tropical hypersurfaces and defect relation

Cao,

Zheng

2018

Preprint

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The tropical Nevanlinna theory is Nevanlinna theory for tropical functions or maps over the max-plux semiring by using the approach of complex analysis. The main purpose of this paper is to study the second main theorem with tropical hypersurfaces into tropical projective spaces and give a defect relation which can be regarded as a tropical version of the Shiffman's conjecture. On the one hand, our second main theorem improves and extends the tropical Cartan's second main theorem due to Korhonen and Tohge [Advances Math. 298(2016), 693-725]. The growth of tropical holomorphic curve is also improved to lim sup r→∞ log T f (r) r = 0 (rather than just hyperorder strictly less than one) by obtaining an improvement of tropical logarithmic derivative lemma. On the other hand, we obtain a new version of tropical Nevanlinna's second main theorem which is different from the tropical Nevanlinna's second main theorem obtained by Laine and Tohge [Proc. London Math. Soc. 102(2011), 883-922]. The new version of the tropical Nevanlinna's second main theorem implies an interesting defect relation that δ f (a) = 0 holds for a nonconstant tropical meromorphic function f with lim sup r→∞ log T f (r) r = 0 and any a ∈ R such that f ⊕ a ≡ a.

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Section: Introductionmentioning

confidence: 99%