We give an introduction to arithmetic dynamics and the questions in the area intended for a broad audience with an emphasis on connections to other branches of mathematics.

We consider the algebraic closure of K((x)), x = (x1, ..., xr), char(K) = 0, namely what we call the field of algebroid Puiseux series, viewed as a subfield of the so-called field of rational polyhedral Puiseux series. Our target is to solve the following problems:

- given a polynomial equation P(x, y) = 0 for P ∈ K[[x]][y], provide a closed form formula for the coefficients of an algebroid...

A univariate sequence is called holonomic, if it satisfies a linear difference equation with polyonomial coefficients. Likewise, a univariate holonomic function satisfies a linear differential equation with polynomial coefficients. In the multivariate (mixed) case, holonomic objects are also characterized through systems of linear difference-differential equations. These equations give a way...

We give an introduction to arithmetic dynamics and the questions in the area intended for a broad audience with an emphasis on connections to other branches of mathematics.

Determining the absolute differential Galois group of interesting differential fields is a central problem in differential Galois theory. For the fields of formal and convergent Laurent series the solution is well-known, but the classical case of rational functions has long resisted a solution. Matzat's conjecture predicts the structure of the absolute differential Galois group of the rational...

Given a partial differential equation (PDE), its solutions can be difficult, if not impossible, to describe.

The purpose of the Fundamental theorem of tropical (partial) differential algebraic geometry is to extract from the equations certain properties of the solutions.

More precisely, this theorem proves that the support of the solutions in $k[[t_1, \cdots, t_m]]$ (with $k$ a field of...

We give an introduction to arithmetic dynamics and the questions in the area intended for a broad audience with an emphasis on connections to other branches of mathematics.

Even though diagonals of multivariate rational functions have been studied from various viewpoints, they still remain quite mysterious objects. An example for this is the widely open conjecture by Christol which characterizes diagonals inside the class of all D-finite functions. In 2012 Bostan, Boukraa, Christol, Hassani, and Maillard created a list with 116 potential counter examples for this...