Holomorphy Rings Research Articles

Density of composite places in function fields and applications to real holomorphy rings

AbstractGiven an algebraic function field and a place ℘ on K, we prove that the places that are composite with extensions of ℘ 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 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 analog to a certain refinement of Artin's solution of Hilbert's 17th problem. We also determine the relation between the topological space of all ‐places of F (places with residue field contained in ), its subspace of all ‐places of F that are composite with the natural ‐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 has only finitely many topological components.

A characterization of finite abelian groups via sets of lengths in transfer Krull monoids

ABSTRACTLet H be a transfer Krull monoid over a finite abelian group G (for example, rings of integers, holomorphy rings in algebraic function fields, and regular congruence monoids in these domains). Then each nonunit a∈H can be written as a product of irreducible elements, say , and the number of factors k is called the length of the factorization. The set L(a) of all possible factorization lengths is the set of lengths of a. It is classical that the system ℒ(H) = {L(a)∣a∈H} of all sets of lengths depends only on the group G, and a standing conjecture states that conversely the system ℒ(H) is characteristic for the group G. Let H′ be a further transfer Krull monoid over a finite abelian group G′ and suppose that ℒ(H) = ℒ(H′). We prove that, if with r≤n−3 or (r≥n−1≥2 and n is a prime power), then G and G′ are isomorphic.

The set of distances in seminormal weakly Krull monoids

The set of distances of a monoid or of a domain is the set of all d∈N with the following property: there are irreducible elements u1,…,uk,v1,…,vk+d such that u1⋅…⋅uk=v1⋅…⋅vk+d, but u1⋅…⋅uk cannot be written as a product of l irreducible elements for any l with k<l<k+d. We show that the set of distances is an interval for certain seminormal weakly Krull monoids which include seminormal orders in holomorphy rings of global fields.

On the density of coprime m-tuples over holomorphy rings

Let [Formula: see text] be a finite field, [Formula: see text] be a function field of genus [Formula: see text] having full constant field [Formula: see text], [Formula: see text] a set of places of [Formula: see text] and [Formula: see text] the holomorphy ring of [Formula: see text]. In this paper, we compute the density of coprime [Formula: see text]-tuples of elements of [Formula: see text]. As a side result, we obtain that whenever the complement of [Formula: see text] is finite, the computation of the density can be reduced to the computation of the [Formula: see text]-polynomial of the function field. In the genus zero case, classical results for the density of coprime [Formula: see text]-tuples of polynomials are obtained as corollaries.

The set of minimal distances in Krull monoids

Let $H$ be a Krull monoid with finite class group $G$. Then every non-unit $a \in H$ can be written as a finite product of atoms, say $a=u_1 \cdot \ldots \cdot u_k$. The set $\mathsf L (a)$ of all possible factorization lengths $k$ is called the set of lengths of $a$. If $G$ is finite, then there is a constant $M \in \mathbb N$ such that all sets of lengths are almost arithmetical multiprogressions with bound $M$ and with difference $d \in \Delta^* (H)$, where $\Delta^* (H)$ denotes the set of minimal distances of $H$. We show that $\max \Delta^* (H) \le \max \{\exp (G)-2, \mathsf r (G)-1\}$ and that equality holds if every class of $G$ contains a prime divisor, which holds true for holomorphy rings in global fields.

Arithmetic of seminormal weakly Krull monoids and domains

We study the arithmetic of seminormal v-noetherian weakly Krull monoids with nontrivial conductor which have finite class group and prime divisors in all classes. These monoids include seminormal orders in holomorphy rings in global fields. The crucial property of seminormality allows us to give precise arithmetical results analogous to the well-known results for Krull monoids having finite class group and prime divisors in each class. This allows us to show, for example, that unions of sets of lengths are intervals and to provide a characterization of half-factoriality.

THE CATENARY DEGREE OF KRULL MONOIDS II

Let$H$be a Krull monoid with finite class group$G$such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree$\mathsf{c}(H)$of$H$is the smallest integer$N$with the following property: for each$a\in H$and each pair of factorizations$z,z^{\prime }$of$a$, there exist factorizations$z=z_{0},\dots ,z_{k}=z^{\prime }$of$a$such that, for each$i\in [1,k]$,$z_{i}$arises from$z_{i-1}$by replacing at most$N$atoms from$z_{i-1}$by at most$N$new atoms. To exclude trivial cases, suppose that$|G|\geq 3$. Then the catenary degree depends only on the class group$G$and we have$\mathsf{c}(H)\in [3,\mathsf{D}(G)]$, where$\mathsf{D}(G)$denotes the Davenport constant of$G$. The cases when$\mathsf{c}(H)\in \{3,4,\mathsf{D}(G)\}$have been previously characterized (see Theorem A). Based on a characterization of the catenary degree determined in the paper by Geroldingeret al.[‘The catenary degree of Krull monoids I’,J. Théor. Nombres Bordeaux23(2011), 137–169], we determine the class groups satisfying$\mathsf{c}(H)=\mathsf{D}(G)-1$. Apart from the extremal cases mentioned, the precise value of$\mathsf{c}(H)$is known for no further class groups.

Sets of lengths in maximal orders in central simple algebras

Let O be a holomorphy ring in a global field K, and R a classical maximal O-order in a central simple algebra over K. We study sets of lengths of factorizations of cancellative elements of R into atoms (irreducibles). In a large majority of cases there exists a transfer homomorphism to a monoid of zero-sum sequences over a ray class group of O, which implies that all the structural finiteness results for sets of lengths—valid for commutative Krull monoids with finite class group—hold also true for R. If O is the ring of algebraic integers of a number field K, we prove that in the remaining cases no such transfer homomorphism can exist and that several invariants dealing with sets of lengths are infinite.

Efficient computation of maximal orders in Artin–Schreier extensions

We describe an algorithm, linear in the degree of the field, for computing pseudo bases for integral closures of holomorphy rings in Artin–Schreier extensions of global function fields and a similar algorithm, also linear in the degree of the field, for computing pseudo bases for S-maximal orders of Artin–Schreier extensions of global function fields. We give examples comparing the running time of our algorithm to that of Round 2 and Fraatz (2005).

The set of distances in Krull monoids

Let $H$ be a Krull monoid with finite class group $G$ such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree $\mathsf c (H)$ of $H$ is the smallest integer $N$ with the following property: for each $a \in H$ and each two factorizations $z, z'$ of $a$, there exist factorizations $z = z_0, ..., z_k = z'$ of $a$ such that, for each $i \in [1, k]$, $z_i$ arises from $z_{i-1}$ by replacing at most $N$ atoms from $z_{i-1}$ by at most $N$ new atoms. Under a very mild condition on the Davenport constant of $G$, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between $\mathsf c (H)$ and the set of distances of $H$ and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on $\mathsf c(H)$ and characterize when $\mathsf c(H)\leq 4$.

Rings of square stable range one

A ring S is said to have square stable range one (written ssr ( S ) = 1 ) if a S + b S = S implies that a 2 + b x is a unit for some x ∈ S . In the commutative case, this extends the class of rings of stable range 1, and allows many new examples such as rings of real-valued continuous functions, and real holomorphy rings. On the other hand, ssr ( S ) = 1 sometimes forces S to have stable range 1. For instance, this is the case for exchange rings S, for which ssr ( S ) = 1 is characterized by S / rad S being reduced (or abelian, or right quasi-duo). We also characterize rings S whose (von Neumann) regular elements are strongly regular, by using an element-wise notion of square stable range one. Extending a result of Estes and Ohm, we show that a possibly noncommutative infinite domain with stable range one or square stable range one must have a non-artinian group of units.

The catenary degree of Krull monoids I

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor (for example, a ring of integers in an algebraic number field or a holomorphy ring in an algebraic function field). The catenary degree c(H) of H is the smallest integer N with the following property: for each a 2 H and each two factorizations z, z 0 of a, there exist factorizations z = z0, . . . , zk = z 0 of a such that, for each i 2 (1, k), zi arises from zi−1 by replacing at most N atoms from zi−1 by at most N new atoms. Under a very mild condition on the Davenport constant of G, we establish a new and simple characterization of the catenary degree. This characterization gives a new structural understanding of the catenary degree. In particular, it clarifies the relationship between c(H) and the set of distances of H and opens the way towards obtaining more detailed results on the catenary degree. As first applications, we give a new upper bound on c(H) and characterize when c(H) � 4.

Diophantine Undecidability of Holomorphy Rings of Function Fields of Characteristic 0

Let K be a one-variable function field over a field of constants of characteristic 0. Let R be a holomorphy subring of K, not equal to K. We prove the following undecidability results for R: if K is recursive, then Hilbert’s Tenth Problem is undecidable in R. In general, there exist x 1 ,...,x n ∈R such that there is no algorithm to tell whether a polynomial equation with coefficients in ℚ(x 1 ,...,x n ) has solutions in R.

Powers of totally positive subgroups of fields

For a formally real field F, a totally positive co-torsion subgroup S of F × and m ⩾ 1 , we prove that ( ∑ S ) m ⊆ ∑ S m . This extends a result of Becker. We prove several refinements of this fact related to higher Pythagoras numbers, real holomorphy rings, and sums with mixed powers.

First-order definitions of rational functions and [formula omitted]-integers over holomorphy rings of algebraic functions of characteristic 0

We consider the problem of constructing first-order definitions in the language of rings of holomorphy rings of one-variable function fields of characteristic 0 in their integral closures in finite extensions of their fraction fields and in bigger holomorphy subrings of their fraction fields. This line of questions is motivated by similar existential definability results over global fields and related questions of Diophantine decidability.

The Artin-Lang property for normal real analytic surfaces

We solve the 17th Hilbert Problem and prove the Artin-Lang property for normal real analytic surfaces. Then we deduce that the absolute (resp. relative) holomorphy ring of such a surface consists of all bounded (resp. locally bounded) meromorphic functions.

Real Prüfer extensions of commutative rings

Let A⊂R be an extension of commutative rings with 1. We show that A is totally real (i.e. all maximal ideals of A are real) and A⊂R is a Prufer extension if and only if R is totally real and the holomorphy ring H(R/A) of R over A is A.

Bezout domains with stable range 1

It is shown that certain classes of Bezout domains have stable range 1, and thus are elementary divisor rings. Included is a strengthening of Roquette's principal ideal theorem which states that the holomorphy ring of a family S of valuation rings of a field K, with S having bounded residue fields, is Bezout. A counterpart is also given where a bound is placed on the ramification indices instead of the residue fields, and these results are applied to rings of integer-valued rational functions over these rings. Along the way, characterizations are given of Prüfer domains with torsion class group, Bezout domains, and Bezout domains with stable range 1 in terms of a family { B(t) | t∈K} of numerical semigroups associated with the ring R, and a related family { D(t) | t∈K} of numerical semigroups.

Anneaux d'holomorphie et Positivstellensatz archimédien

In this paper we obtain some new Positivstellensatz for rings satisfying some archimedean properties. We also study the properties of the real holomorphy ring introduced recently by Becker and Powers. An application of these results in functional analysis is the resolution of the moment problem for some compact sets in R n .

Existentially closed fields with holomorphy rings

In this paper we show that the theory of fields together with an integrally closed subring, the theory of formally real fields with a real holomorphy ring and the theory of formally \(p\)-adic fields with a \(p\)-adic holomorphy ring have no model companions in the language of fields augmented by a unary predicate for the corresponding ring.