Krull - Tropical Hypersurfaces

The concepts of tropical semiring and tropical hypersurface, are extended to the case of an arbitrary ordered group. Then, we define the tropicalization of a polynomial with coefficients in a Krull-valued field. After a close study of the properties of the operator “tropicalization” we conclude with an extension of Kapranov’s theorem to algebraically closed fields together with a valuation over an ordered group. Introduction The tropical semi-ring is the set T := R∪ {∞} together with the operations a⊕ b := min{a, b} and a b := a+ b. A tropical hypersurface is a subset of R defined by a polynomial with coefficients in T. A valuation of a field into the real numbers is used to tropicalize algebraic geometry propositions. A naturally real-valued algebraically closed field is the field of Puiseux series. Let K be an algebraically closed real-valued field. In [2] M. Einsieder, M. Kapranov and D. Lind show that the image of an algebraic hypersurface via a valuation into the reals coincides with the non-linearity locus of its tropical map. Valuations into the real numbers are just a particular type of valuations called classical (see for example [8]). In 1932 W. Krull extended the classical definition considering valuations with values in an arbitrary ordered group [7]. Krull’s definition is the one currently used in most articles and reference texts (see for example [12, 3, 11]). ∗MSC:12.70,14B99.