An Ultradiscrete Qrt Mapping from Tropical Elliptic Curves

Recently the area of tropical geometry has introduced the concept of the tropical elliptic group law associated with a tropical elliptic curve. This gives rise to a notion of the tropical QRT mapping. We compute the explicit tropically birational expressions that define the tropical QRTmapping for some arbitrary set of parameters. We consider this a new integrable ultradiscrete system. This also induces a new notion of tropical elliptic functions, different to the ultradiscretized elliptic functions. Recently the area of tropical geometry has received quite a bit of attention [8]. Much of the work in this direction has been developed in parallel to algebraic geometry [4]. Many classical algebraic geometrical results transfer in some manner to the tropical world. In particular, there has recently been an algebraic geometric interpretation of the group law on an elliptic curve [12]. The QRT systems were introduced in [7]. These mappings are birational mappings which can be described in terms of the group law on an elliptic curve [1]. The algebraic geometric interpretation allows us to characterize such mappings [11]. We use the algebraic geometric interpretation considered in [1, 11] to define an analogous QRT mapping defined in terms of the tropical algebraic geometric framework. We use certain properties to derive explicit tropical birational expressions for the tropical QRT mapping. We consider this system integrable in the sense that the system possesses an invariant defined by the fact that its evolution is restricted to a tropical elliptic curve. The ultradiscretization process has been used to connect integrable cellular automata to integrable difference equations through a limiting process [10]. This limiting process brings a subtraction free difference equation of multiplicative type to an equation over on the semiring S. These mappings have their own sense of integrability defined in terms of ultradiscrete Lax pairs [2] and ultradiscrete singularity confinement [3]. Through this process, one derivation of an ultradiscrete QRT mapping has been given and is considered integrable [5]. These systems described by tropical elliptic curves should be related in some manner to the ultradiscretized systems. In section 1 we introduce the concept of a tropical elliptic curves, lines and the tropical group law. In section 2 we derive an explicit expression for the tropical birational mapping we will call the tropical QRT mapping. This motivates our main result, which is the definition of the tropical elliptic function.