Working With Tropical Meromorphic Functions of One Variable

Abstract

In this paper, we survey and study definitions and properties of tropical polynomials, tropical rational functions and in general, tropical meromorphic functions, emphasizing practical techniques that can really carry out computations. For instance, we introduce maximally represented tropical polynomials and tropical polynomials in compact forms to quickly find roots of given tropical polynomials. We also prove the existence and uniqueness of tropical theorems for meromorphic functions with prescribed roots and poles. Moreover, we explain the relations between classical and tropical meromorphic functions. Different definitions and applications of tropical meromorphic functions are discussed. Finally, we point out the properties of tropical meromorphic functions are very similar to complex ones and prove some tropical analogues of theorems in complex analysis.2010 Mathematics Subject Classification. 14A10,52B20,30C15.

Second main theorem in the tropical projective space

“…Proof. We include an alternative proof to the one given by Tsai [20]. Suppose first that a tropical meromorphic function is given by the formula (3.1).…”

“…These include a generalized ultra-discrete version of Clunie's lemma, and an analogue of Mohon'kos' lemma on value distribution of meromorphic solutions of differential equations. A study of general fundamental properties of tropical meromorphic functions has been performed by Tsai in [20]. Tsai mainly discusses the family of piecewise linear functions defined on the extended real line R ∪ {−∞}, and he calls tropical meromorphic functions defined on R by the name R-tropical meromorphic.…”

Polynomial convolutions in max-plus algebra

“…The coefficients of FCF maxpolynomials form a concave sequence (these polynomials are also called concavified polynomials or polynomials of maximum canonical form [2], maximally represented maxpolynomials [32] or least coefficient minpolynomials in the min-plus setting [15]). As before, we adopt the convention that monomials with coefficient ε are not written out and then it is clear that every monomial is in full canonical form; indeed ax n = a(x ⊕ ε) n .…”

Valuations of semirings

“…Fix a representative g(T ) = b i T i of g(T ). From Lemma 3.2 of [Tsa12], there exists a real number N such that if x > N , then f (x) = a 0 and g(x) = b 0 . Since we know that f (T ) and g(T ) agree on all elements of Q max but −∞, we conclude that f (x) = a 0 = b 0 = g(x) for x > N .…”

“…It is known that, for Q max [T ], the fundamental theorem of tropical algebra holds, i.e., a polynomial P (T ) ∈ Q max [T ] can be uniquely factored into linear polynomials in Q max [T ] (cf [SS09]. or[Tsa12]). In particular, this implies that the notion of the degree of f (T ) ∈ Q max [T ] is well defined.…”