This paper studies notions of scalar relations and crispness of relations.

Let a, a1, . . . , ad be positive integers, m1, . . . ,md nonnegative integers, and z1, . . . , zd complex numbers. We study expressions of the form ∑

These notes record the basic results about DVR’s (discrete valuation rings) and Dedekind rings, with at least sketches of the non-trivial proofs, none of which are hard. This is standard material that any educated mathematician with even a mild interest in number theory should know. It has often slipped through the cracks of Chicago’s first year graduate program, but then we would need at least...

In this paper we provide a constructive version of Tits alternative for a broad class of quaternions with algebraic coefficients. Our result is a generalization of that contained in the paper [1], concerning groups of rational quaternions. Indeed, the tools developed in [1] can be extended to arbitrary number fields by translating them in the corresponding Dedekind domain, as the techniques inv...

We define certain higher-dimensional Dedekind sums that generalize the classical Dedekind-Rademacher sums, and show how to compute them effectively using a generalization of the continued-fraction algorithm. We present two applications. First, we show how to express special values of partial zeta functions associated to totally real number fields in terms of these sums via the Eisenstein cocycl...

An interesting problem in the theory of quasi-groups is to determine how strong an associative law must be assumed in order to obtain a theory of normal subquasi-groups similar to that of ordinary groups. The properties of normal subgroups which it is desirable to retain in the non-associative case are, first, that they form a Dedekind structure, and second, that each one gives rise to a quotie...

We study higher-dimensional analogs of the Dedekind–Carlitz polynomials c (u, v; a, b) := b−1 X

Tits has defined Kac–Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley–Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac–Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis–Tits presentation of the finite groups of Lie type. Whe...

We give a practical criterion characterizing the monogenicity of the integral closure of a Dedekind ring R, based on results on the resultant Res(P,Pi) of the minimal polynomial P of a primitive integral element and of its irreducible factors Pi modulo prime ideals of R. We obtain a generalization and an improvement of the Dedekind criterion (Cohen, 1996) and we give some applications in the ca...