Second main theorem with tropical hypersurfaces and defect relation

Abstract

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.

Second main theorem in the tropical projective space

“…We will also examine some properties of the defect as a real valued function. Finally we will disprove the tropical version of Griffiths conjecture [8] which was proposed by Cao and Zheng [2,Conjecture 4.11]. Halburd and Southall [9] described the following method for generating continuous piecewise linear solutions for ultra-discrete equations of the type (2.1)…”

“…Korhonen and Tohge [12] extended tropical Nevanlinna theory for n-dimensional tropical projective space and proved a tropical analogue of Cartan's second main theorem, which implies the second main theorem by Laine and Tohge. Cao and Zheng [2] further extended tropical Nevanlinna theory for n-dimensional tropical hypersurfaces, and weakened the growth condition of the tropical analogue of the lemma on the logarithmic derivative from functions with hyper-order less than 1 to subnormal functions. Their improvement of the growth condition in the tropical analogue of the lemma on the logarithmic derivative directly implies a corresponding improvement in the growth conditions of the tropical second main theorem and many other results that rely on the lemma on the logarithmic derivative.…”

Logarithmic difference lemma in several complex variables and partial difference equations

“…Here, we improve and extend the known results on logarithmic difference lemma directly for meromorphic fucntions of one and several complex variables by using a growth lemma for nondecreasing positive logarithmic convex function due to Zheng-Korhonen, but avoiding the subharmonic function theory. A tropical version of logarithmic derivative lemma due to Cao and Zheng [4] was obtained very recently.…”

“…If using the Hinkkanen's Borel type Growth Lemma but not Lemma 2.1, we can obtain another form of the logarithmic difference lemma as follows. A tropical version is also given by Cao and Zheng [4] at the same time.…”

“…Let c ∈ C m \{0}. Motivated by the ideas of [4,39], we will prove an improvement of the logarithmic difference lemma in several complex variables [24,2]…”