arXiv subjects

Katarzyna Kuhlmann

Publications and source records attributed to Katarzyna Kuhlmann.

9 recordsLinked to original sources

arXiv2025

Arithmetic of cuts in ordered abelian groups and of ideals over valuation rings

We investigate existence, uniqueness and maximality of solutions $T$ for equations $S_1+T=S_2$ and inequalities $S_1+T\subseteq S_2$ where $S_1$ and $S_2$ are final segments of ordered abelian groups. Since cuts are determined by their upper cut sets, which are final segments, this gives information about the corresponding equalities and inequalities for cuts. We apply our results to investigate existence, uniqueness and maximality of solutions $J$ for equations $I_1 J=I_2$ and inequalities $I_1 J\subseteq I_2$ where $I_1$ and $I_2$ are ideals of valuation rings. This enables us to compute the annihilators of quotients of the form $I_1/I_2\,$.

math.AC↗

arXiv2025

Algorithmic Construction of Real Hyperfields from Minimal Axioms

We study real hyperfields, focusing in particular on those that are finite with cyclic positive cones. All real hyperfields have characteristic zero, although they can still be classified using the C-characteristic, an invariant that captures essential structural information. We present an algorithm to determine all such hyperfields up to isomorphism and compute their C-characteristic. The algorithm is optimal in the sense that the set of axioms used is minimal. We develop and implement this algorithm in software, enabling a complete classification of finite real hyperfields with cyclic positive cones of order up to 15, as well as identification of the C-characteristic that occur in such hyperfields of order up to 17. Restricting attention to finite hyperfields of cyclic positive cones enables substantial simplification of the algorithm, thereby enhancing its computational efficiency and allowing for the rapid generation of hyperfields of large order. Using a criterion that allows us to determine whether a given finite real hyperfield is a Krasner quotient hyperfield, we obtain many new examples of hyperfields that do not arise from Krasner's quotient construction.

math.RA↗

arXiv2021

Orderings and valuations in hyperfields

We introduce and study in detail the notion of compatibility between valuations and orderings in real hyperfields. We investigate their relation with valuations and orderings induced on factor and residue hyperfields. Much of the theory from real fields can be generalized to real hyperfields; we point out facts that cannot. We generalize the Baer-Krull theorem to real hyperfields.

math.AC↗

arXiv2021

Density of Composite Places in Function Fields and Applications to Real Holomorphy Rings

Given an algebraic function field $F|K$ and a place $\wp$ on $K$, we prove that the places that are composite with extensions of $\wp$ to finite extensions of $K$ lie dense in the space of all places of $F$, in a strong sense. We apply the result to the case of $K=R$ any real closed field and the fixed place on $R$ being its natural (finest) real place. This leads to a new description of the real holomorphy ring of $F$ which can be seen as an analogue to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space $M(F)$ of all $\R$-places of $F$ (places with residue field contained in $\R$), its subspace of all $\R$-places of $F$ that are composite with the natural $\R$-place of $R$, and the topological space of all $R$-rational places. Further results about these spaces as well as various classes of relative real holomorphy rings are proven. At the conclusion of the paper the theory of real spectra of rings will be applied to interpret basic concepts from that angle and to show that the space $M(F)$ has only finitely many topological components.

math.AC↗

arXiv2020

A generic approach to measuring the strength of completeness/compactness of various types of spaces and ordered structures

With a simple generic approach, we develop a classification that encodes and measures the strength of completeness (or compactness) properties in various types of spaces and ordered structures. The approach also allows us to encode notions of functions being contractive in these spaces and structures. As a sample of possible applications we discuss metric spaces, ultrametric spaces, ordered groups and fields, topological spaces, partially ordered sets, and lattices. We describe several notions of completeness in these spaces and structures and determine their respective strengths. In order to illustrate some consequences of the levels of strength, we give examples of generic fixed point theorems which then can be specialized to theorems in various applications which work with contracting functions and some completeness property of the underlying space. Ball spaces are nonempty sets of nonempty subsets of a given set. They are called spherically complete if every chain of balls has a nonempty intersection. This is all that is needed for the encoding of completeness notions. We discuss operations on the sets of balls to determine when they lead to larger sets of balls; if so, then the properties of the so obtained new ball spaces are determined. The operations can lead to increased level of strength, or to ball spaces of newly constructed structures, such as products. Further, the general framework makes it possible to transfer concepts and approaches from one application to the other; as examples we discuss theorems analogous to the Knaster--Tarski Fixed Point Theorem for lattices and theorems analogous to the Tychonoff Theorem for topological spaces.

math.GN↗

arXiv2018

Construction of ball spaces and the notion of continuity

Spherically complete ball spaces provide a framework for the proof of generic fixed point theorems. For the purpose of their application it is important to have methods for the construction of new spherically complete ball spaces from given ones. Given various ball spaces on the same underlying set, we discuss the construction of new ball spaces through set theoretic operations on the balls. A definition of continuity for functions on ball spaces leads to the notion of quotient spaces. Further, we show the existence of products and coproducts and use this to derive a topological category associated with ball spaces.

math.GN↗

arXiv2016

Places, cuts and orderings of function fields

In this paper we investigate the space of $\mathbb{R}$-places of an algebraic function field of one variable. We deal with the problem of determining when two orderings of such a field correspond to a single $\mathbb{R}$-place. To this end we introduce and study the space of cuts on a real curve and prove that the space is homeomorphic to the space of orderings. Finally, we prove that two cuts (consequently, two orderings) correspond to a single $\mathbb{R}$-place if they are induced by a single ultrametric ball.

math.AG↗

arXiv2013

A common generalization of metric, ultrametric and topological fixed point theorems - alternative version

We present a general fixed point theorem which can be seen as the quintessence of the principles of proof for Banach's Fixed Point Theorem, ultrametric and certain topological fixed point theorems. It works in a minimal setting, not involving any metrics. We demonstrate its applications to the metric, ultrametric and topological cases, and to ordered abelian groups and fields.

math.AC↗

arXiv2013

Embedding theorems for spaces of $\R$-places of rational function fields and their products

We study spaces $M(R(y))$ of $\R$-places of rational function fields $R(y)$ in one variable. For extensions $F|R$ of formally real fields, with $R$ real closed and satisfying a natural condition, we find embeddings of $M(R(y))$ in $M(F(y))$ and prove uniqueness results. Further, we study embeddings of products of spaces of the form $M(F(y))$ in spaces of $\R$-places of rational function fields in several variables. Our results uncover rather unexpected obstacles to a positive solution of the open question whether the torus can be realized as a space of $\R$-places.

math.AC↗