Tropical plane geometric constructions: a transfer technique in Tropical Geometry

Abstract

The notion of geometric construction is introduced. This notion allows to compare incidence configurations both lying in the algebraic and the tropical plane. We provide sufficient conditions in a geometric construction to ensure that there is always an algebraic counterpart related by tropicalization. We also present some results to detect if this algebraic counterpart cannot exist. With these tools, geometric constructions are applied to transfer classical theorems to the tropical framework, we provide a notion of "constructible incidence theorem" and then several tropical versions of classical theorems are proved such as the converse of Pascal's, Fano's or Cayley-Bacharach theorems.