An ultradiscrete integrable map arising from a pair of tropical elliptic pencils

TL;DR

This paper introduces a tropical geometric framework for an ultradiscrete integrable map with a concave polygon invariant curve, utilizing tropical elliptic curves and pencils, expanding the scope of tropical geometry applications.

Contribution

It provides a novel tropical geometric description of an ultradiscrete map with a concave invariant curve, linking tropical elliptic curves and ultradiscrete QRT maps.

Findings

The map's invariant curve is a concave polygon.

The description uses the addition formula of tropical elliptic curves.

The map arises from a pair of tropical elliptic pencils.

Abstract

We present a tropical geometric description of a piecewise linear map whose invariant curve is a concave polygon. In contrast to convex polygons, a concave one is not directly related to tropical geometry; nevertheless the description is given in terms of the addition formula of a tropical elliptic curve. We show that the map is arising from a pair of tropical elliptic pencils each member of which is the invariant curve of the ultradiscrete QRT map.