Diane Maclagan , Bernd Sturmfels : “ Introduction to Tropical Ge -

What is tropical geometry about? Back in 2005 an influential paper by Richter-Gebert, Sturmfels and Theobald [11] answered that question in the following way: “Tropical algebraic geometry is the geometry of the tropical semiring (R,min,+). Its objects are polyhedral cell complexes which behave like complex algebraic varieties.” Let us look at plane algebraic curves and their tropicalizations to get an idea how this works. To this end consider a plane algebraic curve C, which arises as the vanishing locus of a single bivariate polynomial f (over an algebraically closed field K of characteristic zero). Instead of picking the complex numbers for K, however, here it is more rewarding to take the field of formal Puiseux series with complex coefficients. These are the formal power series with rational exponents which share a common denominator. Puiseux series have been used for the resolution of singularities. As their special feature they admit a non-trivial valuation by sending Puiseux series to their smallest exponents. The tropicalization of the algebraic curve C now arises from applying the valuation map to C pointwise and coordinatewise. One then defines the tropical plane curve T (C) as the topological closure of the image of the valuation map in R. The tropical curve T (C) is an unbounded one-dimensional polyhedral complex, equipped with integral weights, which still “knows” a lot about the original curve C. For instance, from T (C) one can see the Newton polygon and thus the degree of C. Moreover, the dimension of the space of cycles of T (C), seen as a planar graph, equals the arithmetic genus of C. Again by employing the valuation map, one can also tropicalize the polynomial f which defines the algebraic curve C. It is an essential feature that, via polyhedral combinatorics, one can obtain the tropical curve T (C) also directly from the tropicalization of f , which is a polynomial over the tropical semi-ring (R,min,+); see [6]. To explain the concept let us look at the quadratic bivariate polynomial f(x, y) = t+ x− tx + y + ty with coefficients in the field C{{t}} of complex Puiseux series, which are power series in t with rational exponents. The curve C defined by f is a conic. To visualize the shape of C it is helpful, for a brief moment, to think of t as a small positive real number. Then f would become a bivariate real polynomial whose real locus is a hyperbola. However, we return to computing with Puiseux series. For any given ξ ∈ C{{t}} we can solve the equation f(ξ, η) = 0, and we arrive at