An algebraic method for quasi-linear first-order ODEs

In this paper, we consider the class of quasi-linear first-order ODEs of the form y′ = P(x; y), where P is a polynomial in y with coefficients in ℂ(x), and study their algebraic solutions. Our method is intrinsically based on algebraic function field theory. We give an upper bound for the degrees of algebraic solutions which are in a quadratic extension of ℂ(x).

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.

THE STABILIZERS IN A DRINFELD MODULAR GROUP OF THE VERTICES OF ITS BRUHAT–TITS TREE: AN ELEMENTARY APPROACH

Let K be an algebraic function field of one variable with constant field k and let [Formula: see text] be the Dedekind domain consisting of all those elements of K which are integral outside a fixed place ∞ of K. When k is finite the group [Formula: see text] plays a central role in the theory of Drinfeld modular curves analogous to that played by SL2(ℤ) in the classical theory of modular forms. When k is finite (respectively, infinite) we refer to a group [Formula: see text] as an arithmetic (respectively, non-arithmetic) Drinfeld modular group. Associated with [Formula: see text] is its Bruhat–Tits tree, [Formula: see text]. The structure of the group is derived from that of the quotient graph [Formula: see text]. Using an elementary approach which refers explicitly to matrices we determine the structure of all the vertex stabilizers of [Formula: see text]. This extends results of Serre, Takahashi and the authors. We also determine all possible valencies of the vertices of [Formula: see text] for the important special case where ∞ has degree 1.

FREE GROUP ALGEBRAS IN DIVISION RINGS

Let k be an algebraically closed field of characteristic zero and let L be an algebraic function field over k. Let σ : L → L be a k-automorphism of infinite order, and let D be the skew field of fractions of the skew polynomial ring L[t; σ]. We show that D contains the group algebra kF of the free group F of rank 2.

DIOPHANTINE DEFINABILITY OVER HOLOMORPHY RINGS OF ALGEBRAIC FUNCTION FIELDS WITH INFINITE NUMBER OF PRIMES ALLOWED AS POLES

Let K be an algebraic function field over a finite field of constants of characteristic greater than 2. Let W be a set of non-archimedean primes of K, let [Formula: see text]. Then for any finite set S of primes of K there exists an infinite set W of primes of K containing S, with the property that OK,S has a Diophantine definition over OK,W.

Linear Dynamical Systems of Higher Genus

A class of linear systems which after ordinary linear systems are in a certain sense the simplest ones, is associated with every algebraic function field. From the standpoint developed in the paper ordinary linear systems are associated with the rational function field.