Rev. Mat. Iberoamericana 27(2011), no.1, 181…232
                                    Tropical plane geometric
                            constructions: a transfer technique
                                      in Tropical Geometry
                                                                     Luis Felipe Tabera
                                                   Abstract
                              The notion of geometric construction is introduced. This notion
                           allows to compare incidence con“gurations both lying in the algebraic
                           andthetropical plane. We providesufficient 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 theoremŽ
                           and then several tropical versions of classical theorems are proved
                           such as the converse of Pascals, Fanos or Cayley-Bacharach theo-
                           rems.
                      1. Introduction
                      Let K be an algebraically closed “eld provided with a nontrivial rank one
                      valuation v and valuation group T. We suppose that T is a subgroup of the
                                ∗
                      reals, v : K Š→ T ⊆ R. We have the following map on the algebraic torus:
                                            ∗ n                  n
                                   T :(K)          Š→           T
                                       (x ,...,x )  → (Šv(x ),...,Šv(x ))
                                         1      n            1          n
                      This mapis the tropicalization or projection map. Tropical varieties are then
                                                                  ∗ n
                      de“ned as the image of an algebraic variety V ⊆ (K ) under the tropicaliza-
                      tion map T. One of the most interesting aspects of tropical varieties is that
                      they inherit relevant geometric properties from their algebraic counterparts.
                      In the present work we explore this inheritance for the case of translating
                      2000 Mathematics Subject Classification: Primary: 14H99, Secondary: 51E30, 12J25,
                      16Y60.
                      Keywords: Tropical geometry, geometric constructions, incidence con“gurations.
           182 L.F. Tabera
           incidence theorems of classical projective geometry to the tropical context.
           The origin of this work is the Pappus theorem counterexample in [11]. In
           that paper, a tropical con“guration of points and lines in the shape of Pap-
           pus theorem hypotheses is shown such that it does not verify Pappus thesis.
           In particular, it implies that this con“guration is not the projection of a
           similar con“guration of points and lines in the algebraic plane. The authors
           provided then another alternative version of the same theorem and claimed
           that this new version would hold in the tropical context. The key of this
           newversion of Pappus theorem is that the hypotheses are given as the result
           of a geometric construction dealing with points and lines. The correctness
           of this theorem was shown in [13] using some precursor techniques on geo-
           metric constructions. Following this idea, many incidence theorems can be
           given as a construction of a con“guration of curves and points (hypothesis)
           and then some information is derived (the thesis of the theorem). Thus, we
           will focus on geometric constructions in the plane and how they behave with
           respect to tropicalization.
             Intuitively, a geometric construction is a procedure that starts with a
           set of input curves and points and then de“nes other curves and points by
           either intersecting two available curves or computing a curve de“ned by a
           polynomial of “xed support passing through a set of points (a conic through
           “ve points, for example). The main algorithm we present consists in tak-
           ing a tropical instance of a geometric construction and then computing a
           constructible set S, over the residual “eld of the valuation, that encodes
           sufficient conditions for the compatibility with tropicalization of an alge-
           braic geometric construction. We will also show some certi“cates during the
           computation to detect if a tropical realization of a geometric construction is
           not the projection of any algebraic realization.
             Moreover, we present a notion of admissible geometric construction.This
           is a combinatorial notion that ensures that for all tropical realizations of the
           construction, the computed set S isnonemptyanddense.Thatis,there
           will always be an algebraic preimage of the construction under the tropical-
           ization T. This notion can be applied to prove that some incidence theorems
           (so-called constructible incidence theorems) hold in the tropical context if
           we are able to describe their hypotheses as the output of an admissible
           geometric construction.
             The paper is structured as follows: in Section 2 we present the notion
           of geometric construction and show how to understand the steps of a con-
           struction in both the algebraic and the tropical context. In Section 3 we
           provide the main algorithm of the article. Then, the limits of our geometric
           construction method are shown by a series of examples. Furthermore, we
           include a generalization of the notion of admissibility related with the notion
                                                                  Tropical Plane Geometric Constructions                  183
                                of points in general position with respect to a curve. Finally, in Section 4, we
                                use the results obtained so far to build up a notion of constructible incidence
                                theorem that is compatible with tropicalization and we show some relevant
                                instances of theorems of this kind.
                                1.1. Notation and preliminaries
                                Let k be the residual “eld of K by the valuation. There are three main
                                cases of valued “elds according to the characteristics: the case char(K)=
                                char(k) = 0 (equicharacteristic zero), the case char(K)=char(k)=p>0
                                (positive characteristic) and the case char(K)=0