Second main theorem in the tropical projective space

Korhonen, Risto; Tohge, Kazuya

doi:10.1016/j.aim.2016.05.003

Cited by 4 publications

(10 citation statements)

References 26 publications

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“…From the viewpoint of geometry, we see that a zero (or pole) of f is the point x ∈ R at which the graph of f is nonlinear and convex (or concave). Korhonen and Tohge [15,Proposition 3.3] proved that for any tropical meromorphic function f, there exist two tropical entire functions g and h such that f = h ⊘ g, where g and h do not have any common zeros.…”

Section: Definition 22 [14]mentioning

confidence: 99%

“…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%

“…. , P q • f }) = 0 so that the Korhonen and Tohge's version of tropical Cartan second main theorem [15] is improved. Moreover, from the improvement of tropical Cartan second main theorem, we obtain new versions of tropical Nevanlinna second main theorem (Theorem 6.3 and Theorem 6.5) which are different from the Laine and Tohge's version of tropical second main theorem, and thus propose an interesting defect relation: 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.…”

Section: Introductionmentioning

confidence: 99%

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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: Definition 22 [14]mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Second main theorem with tropical hypersurfaces and defect relation

Cao,

Zheng

2018

Preprint

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“…There are differential projective spaces that finite projective spaces that have applications in analysis and discrete mathematics. Many authors mainly had paid attention to study the projective spaces and their applications, see [10][11][12][13][14][15].…”

Section: Introductionmentioning

confidence: 99%

On the quaternion projective space

Omar

Shahin

Ahmed

et al. 2020

Journal of Taibah University for Science

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Apart from being a vital and exciting field in mathematics with interesting results, projective spaces have various applications in design theory, coding theory, physics, combinatorics, number theory and extremal combinatorial problems. In this paper, we consider real, complex and quaternion projective spaces. We focus on the geometric feature of the sectional curvatures. We first study the real and complex projective spaces. We prove that their sectional curvatures are constant. Then, we consider the quaternion projective space. Specifically, we prove that the quaternion projective space has a positive constant sectional curvature. We also determine the pinching constant for the complex and quaternion projective spaces.

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“…The original paper [5] on tropical Nevanlinna theory in the real line (restricting to integer slopes) has been subsequently extended in [10] to include tropical counterpart to the second main theorem (with arbitrary real slopes). A further extension has been recently made in [8] to considering tropical holomorphic curves in a tropical projective space. In these preceding papers, tropical Nevanlinna theory has been treated in the global setting.…”

Section: Introduction and Notationmentioning

confidence: 99%

Tropical meromorphic functions in a finite interval

Laine¹,

Tohge²

2019

Ann. Acad. Sci. Fenn. Math.

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The tropical Nevanlinna theory in the whole real line R describes value distribution of continuous piecewise linear functions of a real variable with arbitrary real slopes, called tropical meromorphic functions, similarly as value distribution of meromorphic functions of a complex variable is described by the classical Nevanlinna theory in the whole complex plane C. As a tropical counterpart to the Nevanlinna theory in a disc or an annulus centered at the origin, we introduce in this paper a value distribution theory of continuous piecewise linear functions in a symmetric finite open interval (−R, R). The shift operator (difference operator) has a key role in the tropical value distribution theory in R corresponding to the role of the differential operator in the Nevanlinna theory in a subregion of C. However, the affine shift x → x + c does not operate properly in finite intervals. Therefore, we introduce a shift x → s τ (x) which may be called as the tropical hyperbolic shift. This notion enables us to obtain the quotient estimate m r, f s τ (x) ⊘ f (x) = o(1)T (r, f ) for tropical meromorphic functions f (x) defined in an interval (−R, R) in R, corresponding to the logarithmic derivative estimate in the Nevanlinna theory for meromorphic functions f (z) defined in a disc or in an annulus. A sort of the second main theorem is also stated by means of this estimate. Concerning hyperbolic shift and the second main theorem, we assume an order restriction to f (x). This restriction is shown to be necessary by an example.

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Second main theorem in the tropical projective space

Cited by 4 publications

References 26 publications

Second main theorem with tropical hypersurfaces and defect relation

Second main theorem with tropical hypersurfaces and defect relation

On the quaternion projective space

Tropical meromorphic functions in a finite interval

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