Given an algebraically closed field K of characteristic zero, we present the Abhyankar–Jung theorem for any excellent henselian ring whose completion is a formal power series ring K[[z]]. In particular, examples include the local rings which form a Weierstrass system over the field K. The Abhyankar–Jung theorem may be regarded as a higher dimensional counterpart of the Newton–Puiseux theorem. I...

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...

There is a rule of thumb that in 90% of all cases when a proof in algebra or combinatorics seems to use analysis, this use can be easily avoided. For example, when a proof of a combinatorial identity uses power series, it is in most cases enough to replace the words ”power series” by ”formal power series”, and there is no need anymore to worry about issues of convergence and well-definedness. W...

A constructive version of Newton–Puiseux theorem for computing the Puiseux expansions of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field need not be algebraically closed and a factorization algorithm of polynomials over the base field is not needed. The extensions obtained are a type of regula...

A constructive version of Newton–Puiseux theorem for computing the Puiseux expansions of algebraic curves is presented. The proof is based on a classical proof by Abhyankar. Algebraic numbers are evaluated dynamically; hence the base field need not be algebraically closed and a factorization algorithm of polynomials over the base field is not needed. The extensions obtained are a type of regula...

We study the dynamics of polynomials with coefficients in a nonArchimedean field L, where L is the completion of an algebraic closure of the field of formal Laurent series. We prove that every wandering Fatou component is contained in the basin of a periodic orbit. We give a dynamical characterization of polynomials having algebraic Julia sets. More precisely, we establish that a polynomial wit...