referred to simply as the tropical semiring. Conventions differ between authors and subfields: some use the min convention, some use the max convention. The tropical semiring operations model how valuations behave under addition and multiplication in a valued field. Some common valuated fields encountered in tropical geometry (with min convention) are: or with the trivial valuation, for all . or its extensions with the p-adic valuation, for a and b coprime to p. The field of Laurent series (integer powers), or the field of (complex) Puiseux series , with valuation returning the smallest exponent of t appearing in the series. Tropical polynomials A tropical polynomial is a function that can be expressed as the tropical sum of a finite number of monomial terms. A monomial term is a tropical product (and/or quotient) of a constant and variables from . Thus a tropical polynomial F is the minimum of a finite collection of affine-linear functions in which the variables have integer coefficients, so it is concave, continuous, and piecewise linear. Given a polynomial f in the Laurent polynomial ring where K is a valued field, the tropicalization of f, denoted , is the tropical polynomial obtained from f by replacing multiplication and addition by their tropical counterparts and each constant in K by its valuation. That is, if then The set of points where a tropical polynomial F is non-differentiable is called its associated tropical hypersurface, denoted (in analogy to the vanishing set of a polynomial). Equivalently, is the set of points where the minimum among the terms of F is achieved at least twice. When for a Laurent polynomial f, this latter characterization of reflects the fact that at any solution to , the minimum valuation of the terms of f must be achieved at least twice in order for them all to cancel. Tropical varieties Definitions For X an algebraic variety in the algebraic torus , the tropical variety of X or tropicalization of X, denoted , is a subset of that can be defined in several ways. The equivalence of these definitions is referred to as the Fundamental Theorem of Tropical Geometry. Intersection of tropical hypersurfaces Let be the ideal of Laurent polynomials that vanish on X in . Define When X is a hypersurface, its vanishing ideal is a principal ideal generated by a Laurent polynomial f, and the tropical variety is precisely the tropical hypersurface .