Related papers: Valuations in fields of power series

We examine the concept of field in tensor-triangular geometry. We gather examples and discuss possible approaches, while highlighting open problems. As the construction of residue tt-fields remains elusive, we instead produce suitable…

In recent decades, the defect of finite extensions of valued fields has emerged as the main obstacle in several fundamental problems in algebraic geometry such as the local uniformization problem. Hence, it is important to identify…

This paper shows a finiteness property of a divisorial valuation in terms of arcs. First we show that every divisorial valuation over an algebraic variety corresponds to an irreducible closed subset of the arc space. Then we define the…

In this article we further develop the theory of valuation independence and study its relation with classical notions in valuation theory such as immediate and defectless extensions. We use this general theory to settle two open questions…

The purpose of this article is to give an explicit description, in terms of hypergeometric functions over finite fields, of zeta function of a certain type of smooth hypersurfaces that generalizes Dwork family. The point here is that we…

The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an…

We study modular forms of some congruence subgroups. In this paper, we treat the cases level is 2-power, 3-power or 5. Structures of graded rings and many identities of infinite sum or infinite product are given. Theory of rational (1/3,…

This paper studies modular forms of rank four and level one. There are two possiblities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we…

This paper gives a survey on a valuation theoretical approach to local uniformization in positive characteristic, the model theory of valued fields in positive characteristic, and their connection with the valuation theoretical phenomenon…

Classically, Groebner bases are computed by first prescribing a set monomial order. Moss Sweedler suggested an alternative and developed a framework to perform such computations by using valuation rings in place of monomial orders. We build…

We give a criterion for a group homomorphism on a valued abelian group to be surjective and to preserve spherical completeness. We apply this to give a criterion for the existence of integration on a valued differential field. Further, we…

In this short manuscript, we will put some light on the different outcomes when two non-constant meromorphic functions share a value with prescribed weight two.

We present several aspects of the "topology of meromorphic functions", which we conceive as a general theory which includes the topology of holomorphic functions, the topology of pencils on quasi-projective spaces and the topology of…

We provide axiomatization and relative quantifier elimination for valued fields equipped with an automorphism, in residue characteristic zero. Similar results are known under strong assumptions on the interaction between the automorphism…

We prove a new bound on the number of shared values of distinct meromorphic functions on a compact Riemann surface, explain a mistake in a previous paper on this topic, and give a survey of related questions.

We provide upper bounds for the cardinality of the value set of a polynomial map in several variables over a finite field. These bounds generalize earlier bounds for univariate polynomials.

In some previous papers we have obtained characterizations (we called them characterizations of Lambek-Carlitz type) of some class of formal power series. Now, we discuss, investigate and compare these results from a categorial point of…

The main result from this note provides a constructive characterization of the valuative dimension, which bears a strong analogy to Lombardi's constructive characterization of the Krull dimension. While Lombardi's characterization uses the…

In this paper we consider graded ideals in a polynomial ring over a field and ask when such an ideal has the property that all of its powers have a linear resolution. In particular it is shown that all powers of a monomial ideal with…

Gr\"obner bases have been generalized by replacing monomial orders with constructions such as valuations and filtrations. We consider suitable valuations on a rational valuation field $K(x,y)$ and analyze their behavior when restricting to…