Tropical meromorphic functions in a finite interval

Abstract. 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 7→ x+ c does not operate properly in finite intervals. Therefore, we introduce a shift x 7→ sτ (x) which may be called as the tropical hyperbolic shift. This notion enables us to obtain the quotient estimate m (