Related papers: Superrosy fields and valuations

Liouville closed $H$-fields are ordered differential fields whose ordering and derivation interact in a natural way and where every linear differential equation of order $1$ has a nontrivial solution. (The introduction gives a precise…

A graded-division algebra is an algebra graded by a group such that all nonzero homogeneous elements are invertible. This includes division algebras equipped with an arbitrary group grading (including the trivial grading). We show that a…

The Ruled Residue Theorem asserts that given a ruled extension $(K|k,v)$ of valued fields, the residue field extension is also ruled. In this paper we analyse the failure of this theorem when we set $K$ to be algebraic function fields of…

We prove that if K is an infinite stable field whose generic type has weight 1 then K is separably closed. We also obtain partial results when the generic of K has finite weight and when K is strongly stable.

While the natural model-theoretic ranks available in differentially closed fields (of characteristic zero), namely Lascar and Morley rank, are known not to be definable in families of differential varieties; in this note we show that the…

Let Rep(F;K) denote the category of functors from finite dimensional F-vector spaces to K-modules, where F is a field and K is a commutative ring. We prove that, if F is a finite field, and Char F is invertible in K, then the K-linear…

We describe the valuations following infinitely near singular points of a (singular) holomorphic foliation in the complex plane. They appear to be those satisfying a generalization of L'Hopital's rule. With them, we characterize dicritical…

We know that for a finite field $F$, every function on $F$ can be given by a polynomial with coefficients in $F$. What about the converse? i.e. if $R$ is a ring (not necessarily commutative or with unity) such that every function on $R$ can…

We introduce a new real valued invariant for finitely presented groups called residual deficiency. Its main property is the following. Let G be a finitely presented group. If the residual deficiency of G is greater than one, then G has a…

In this article, we study the structure of finitely ramified mixed characteristic valued fields. For any two complete discrete valued fields $K_1$ and $K_2$ of mixed characteristic with perfect residue fields, we show that if the $n$-th…

Admitting a non-trivial $p$-henselian valuation is a weaker assumption on a field than admitting a non-trivial henselian valuation. Unlike henselianity, $p$-henselianity is an elementary property in the language of rings. We are interested…

Let $E$ be a primarily quasilocal field, $M/E$ a finite Galois extension and $D$ a central division $E$-algebra of index divisible by $[M\colon E]$. In addition to the main result of Part I, this part of the paper shows that if the Galois…

A theorem of Dolfi, Herzog, Kaplan, and Lev \cite[Thm.~C]{DHKL} asserts that in a finite group with trivial Fitting subgroup, the size of the soluble residual of the group is bounded from below by a certain power of the group order, and…

In the second section, we introduce hemiring-valued pseudonormed rings and generalize Albert's result which states that every finite-dimensional algebra can be normed. Next, we introduce shrinkable hemirings and prove that dense division…

We show that an algebraic immediate valuation ring extension of characteristic $p>0$ is a filtered union of complete intersection algebras of finite type.

We previously showed that the inverse limit of standard-graded polynomial rings with perfect coefficient field is a polynomial ring, in an uncountable number of variables. In this paper, we show that the same result holds with arbitrary…

We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these…

A residually finite (profinite) group $G$ is just infinite if every non-trivial (closed) normal subgroup of $G$ is of finite index. This paper considers the problem of determining whether a (closed) subgroup $H$ of a just infinite group is…

Let $S$ be an integral domain with field of fractions $F$ and let $A$ be an $F$-algebra having an $S$-stable basis. We prove the existence of an $S$-subalgebra $R$ of $A$ lying over $S$ whose localization with respect to $S$ is $A$ (we call…

The Cremona group of rank n over a field k is the group of birational automorphisms of the n-dimensional projective space over the field k. We study the minimal dimension such that all finite subgroups of the Cremona group have a faithful…