Christopher Frei

On rings of integers generated by their units

We give an affirmative answer to the following question by Jarden and Narkiewicz: Is it true that every number field has a finite extension L such that the ring of integers of L is generated by its units (as a ring)? As a part of the proof, we generalize a theorem by Hinz on power-free values of polynomials in number fields.

COUNTING IMAGINARY QUADRATIC POINTS VIA UNIVERSAL TORSORS, II

We prove Manins conjecture for four singular quartic del Pezzo surfaces over imaginary quadratic number fields, using the universal torsor method.

Additive unit representations in rings over global fields - A survey

We give an overview on recent results concerning additive unit representations. Furthermore the solutions of some open questions are included. We focus on rings of integers in number fields and in function fields of one variable over perfect fields. The central problem is whether and how certain rings are (additively) generated by their units. In the final section we deal with matrix rings over quaternions and over Dedekind domains. Our point of view is number-theoretic whereas we do not discuss the general algebraic background. 1. The unit sum number In 1954, Zelinsky [44] proved that every endomorphism of a vector space V over a division ring D is a sum of two automorphisms, except if D = Z/2Z and dimV = 1. This was motivated by investigations of Dieudonne on Galois theory of simple and semisimple rings [7] and was probably the first result about the additive unit structure of a ring. Using the terminology of Vamos [41], we say that an element r of a ring R (with unity 1, not necessarily commutative) is k-good if r is a sum of exactly k units of R. If every element of R has this property then we call R k-good. By Zelinsky’s result, the endomorphism ring of a vector space with more than two elements is 2-good. Clearly, if R is k-good then it is also l-good for every integer l > k. Indeed, we can write any element of R as r = (r − (l − k) · 1) + (l − k) · 1, and expressing r − (l − k) · 1 as a sum of k units gives a representation of r as a sum of l units. Goldsmith, Pabst and Scott [20] defined the unit sum number u(R) of a ring R to be the minimal integer k such that R is k-good, if such an integer exists. If R is not k-good for any k then we put u(R) := ω if every element of R is a sum of units, and u(R) :=∞ if not. We use the convention k < ω <∞ for all integers k. Clearly, u(R) ≤ ω if and only if R is generated by its units. Here are some examples from [20] and [41]: • u(Z) = ω, 1991 Mathematics Subject Classification. 00-02, 11R27, 16U60.

Non-unique factorization of polynomials over residue class rings of the integers

We investigate non-unique factorization of polynomials in ℤ p n [x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, ℤ p n [x] is atomic. We reduce the question of factoring arbitrary nonzero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of ℤ p n [x] is a direct sum of monoids corresponding to irreducible polynomials in ℤ p [x], and we show that each of these monoids has infinite elasticity. Moreover, for every m ∈ ℕ, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.

Counting imaginary quadratic points via universal torsors

We prove Manins conjecture for four singular quartic del Pezzo surfaces over imaginary quadratic number elds, using the universal torsor method.

Rational points of bounded height on general conic bundle surfaces

A conjecture of Manin predicts the asymptotic distribution of rational points of bounded height on Fano varieties. In this paper we use conic bundles to obtain correct lower bounds for a wide class of surfaces over number fields for which the conjecture is still far from being proved. For example, we obtain the conjectured lower bound of Manins conjecture for any del Pezzo surface whose Picard rank is sufficiently large, or for arbitrary del Pezzo surfaces after possibly an extension of the ground field of small degree.

Sums of units in function fields

Let R be the ring of S-integers of an algebraic function field (in one variable) over a perfect field, where S is finite and not empty. It is shown that for every positive integer N there exist elements of R that can not be written as a sum of at most N units. Moreover, all quadratic global function fields whose rings of integers are generated by their units are determined.

Forms of differing degrees over number fields

Consider a system of polynomials in many variables over the ring of integers of a number field

GENERALISED DIVISOR SUMS OF BINARY FORMS OVER NUMBER FIELDS

K

Average bounds for the

. We prove an asymptotic formula for the number of integral zeros of this system in homogeneously expanding boxes. As a consequence, any smooth and geometrically integral variety