The theory of Λ-rings, in the sense of Grothendieck’s Riemann– Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual algebraic geometry over the ring Z of integers to produce Λ-algebraic geometry. We show that Λ-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than Z and that it has many properties predicted for a...

We prove that Z is diophantine over the ring of algebraic integers in any totally real number field or quadratic extension of a totally real number field.

In this paper we discuss the basic problems of algorithmic algebraic number theory. The emphasis is on aspects that are of interest from a purely mathematical point of view, and practical issues are largely disregarded. We describe what has been done and, more importantly, what remains to be done in the area. We hope to show that the study of algorithms not only increases our understanding of a...

In this work we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2, 3, 4, 6, 8 and 12, which are rotated versions of the lattices 3n , for n = 2, 3, 4, 6, 8 and K12. These algebraic lattices are constructed through twisted canonical homomorphism via ideals of a ring of algebraic integers. Mathematical subject classification: 18B35, 94A15, ...

3 Extensions of rings and fields 14 3.1 Symmetric polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Integral ring extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Prime ideals in Z[x]: elementary classification . . . . . . . . . . . . . . . . . . . . . . 19 3.4 The spectrum of a commutative ring . . . . . . . . . . . . . ...

Until recently, no Salem numbers were known of trace below −1. In this paper we provide several examples of trace −2, including an explicit infinite family. We establish that the minimal degree for a Salem number of trace −2 is 20, and exhibit all Salem numbers of degree 20 and trace −2. Indeed there are just two examples. We also settle the closely-related question of the minimal degree d of a...

Graph eigenvalues are examples of totally real algebraic integers, i.e. roots of real-rooted monic polynomials with integer coefficients. Conversely, the fact that every totally real algebraic integer occurs as an eigenvalue of some finite graph is a deep result, conjectured forty years ago by Hoffman, and proved seventeen years later by Estes. This short paper provides an independent and eleme...

The original meaning of diophantine problems is to find all solutions of equations in integers or rational numbers, and to give a bound for these solutions. One may expand the domain of coefficients and solutions to include algebraic integers, algebraic numbers, polynomials, rational functions, or algebraic functions. In the case of polynomial solutions, one tries to bound their degrees. Inequa...

We prove several results in the theory of fusion categories using product (norm) and sum (trace) Galois conjugates formal codegrees. First, we that finitely-many exist up to equivalence whose global dimension has a fixed norm. Furthermore, with two exceptions, all codegrees spherical square-free norm are rational integers. This implies, three every braided category prime is pointed. The reason ...