The equation y′ = fy in zero residue characteristic

Field of Characteristic Zero

Let K be a p-adic field of characteristic zero. For polynomials A,B ∈ K[x] we consider decompositions A(x)f(x) +B(x)g(x) = 1 of entire functions f, g on K and try to improve an impossibility result due to A. Boutabaa concerning transcendental f, g. Also, an independent proof of a p-adic diophantic statement due to D. N. Clark is given.

On the Equation Y

Chapter 1 Characteristic p methods in characteristic zero via ultraproducts

Characteristic p methods in characteristic zero via ultraproducts

In recent decades, by exploiting the algebraic properties of the Frobenius in positive characteristic, many so-called homological conjectures and intersection conjectures have been established, culminating into the powerful theory of tight closure and big Cohen–Macaulay algebras. In the present article, I give a survey of how these methods also can be applied directly in characteristic zero by ...

Strong resolution of singularities in characteristic zero

Hironaka's spectacular proof of resolution of singularities is built on a multiple and intricate induction argument. It is so involved that only few people could really understand it. The constructive proofs given later by Villamayor, Bierstone-Milman and Encinas-Villamayor presented important steps towards a better understanding of the reasoning. They describe an algorithmic procedure for reso...