第二主定理与热带超曲面及缺陷关系
Second main theorem with tropical hypersurfaces and defect relation
AI总结：
本文提出热带Nevanlinna第二主定理的改进版本，并推导出关于热带亚纯函数的缺陷关系。
摘要：
热带Nevanlinna理论是通过使用复分析方法来研究热带函数或映射在max-plus半环上的Nevanlinna理论。本文的主要目的是研究将热带超曲面引入热带射影空间中的第二主定理，并给出一个可以视为Shiffman猜想热带版本的缺陷关系。一方面，我们的第二主定理改进并扩展了Korhonen和Tohge [Advances Math. 298(2016), 693-725] 所述的热带Cartan第二主定理。通过获得热带对数导数引理的改进，热带全纯曲线的增长性也得到了改进，即$\limsup_{r\rightarrow\infty}\frac{\log T_{f}(r)}{r}=0$（而不是仅仅超阶严格小于一）。另一方面，我们得到了一个新的热带Nevanlinna第二主定理版本，该版本不同于Laine和Tohge [Proc. London Math. Soc. 102(2011), 883-922] 所获得的热带Nevanlinna第二主定理。新的热带Nevanlinna第二主定理暗示了一个有趣的缺陷关系，即对于非常数的热带亚纯函数$f$，当$\limsup_{r\rightarrow\infty}\frac{\log T_{f}(r)}{r}=0$且对于任何$a\in\mathbb{R}$满足$f\oplus a\not\equiv a$时，$δ_{f}(a)=0$成立。
英文摘要：
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 $\limsup_{r\rightarrow\infty}\frac{\log T_{f}(r)}{r}=0$ (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 $\limsup_{r\rightarrow\infty}\frac{\log T_{f}(r)}{r}=0$ and any $a\in\mathbb{R}$ such that $f\oplus a\not\equiv a.$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
