Exact values of Γ∗(10,p)

For [Formula: see text] and [Formula: see text] a prime number, define [Formula: see text] to be the smallest positive integer [Formula: see text] such that any diagonal form [Formula: see text], with integer coefficients, has nontrivial zero over [Formula: see text] whenever [Formula: see text]. A special case of a conjecture attributed to Artin states that [Formula: see text]. It is well known that the equality occurs when [Formula: see text]. In this paper, we obtain the exact values of [Formula: see text] for all primes [Formula: see text] and, except for [Formula: see text], these values are much lower than those established in the conjecture, as might be expected.

A zero-dimensional topologically nontrivial state in a superconducting quantum dot

The classification of topological states of matter in terms of unitary symmetries and dimensionality predicts the existence of nontrivial topological states even in zero-dimensional systems, i.e., systems with a discrete energy spectrum. Here, we show that a quantum dot coupled with two superconducting leads can realize a nontrivial zero-dimensional topological superconductor with broken time-reversal symmetry, which corresponds to the finite size limit of the one-dimensional topological superconductor. Topological phase transitions corresponds to a change of the fermion parity, and to the presence of zero-energy modes and discontinuities in the current–phase relation at zero temperature. These fermion parity transitions therefore can be revealed by the current discontinuities or by a measure of the critical current at low temperatures.

The u-invariant of p-adic function fields

Over a finitely generated field extension in m variables over a p-adic field, any quadratic form in more than 2m + 2 variables has a nontrivial zero. This bound is sharp. We extend this result to a wider class of fields. A key ingredient to our proofs is a recent result of Heath-Brown on systems of quadratic forms over p-adic fields.

NORMAL PAIRS WITH ZERO-DIVISORS

Results of Davis on normal pairs (R, T) of domains are generalized to (commutative) rings with nontrivial zero-divisors, particularly complemented rings. For instance, if T is a ring extension of an almost quasilocal complemented ring R, then (R, T) is a normal pair if and only if there is a prime ideal P of R such that T = R[P], R/P is a valuation domain and PT = P. Examples include sufficient conditions for the "normal pair" property to be stable under formation of infinite products and ⋈ constructions.

Topological Aspects of the Product of Lattices

Let be an arbitrary nonempty set and a lattice of subsets of such that , . () denotes the algebra generated by , and () denotes those nonnegative, finite, finitely additive measures on (). In addition, () denotes the subset of () which consists of the nontrivial zero-one valued measures. The paper gives detailed analysis of products of lattices, their associated Wallman spaces, and products of a variety of measures.

On systems of diagonal forms II

Given a system of diagonal forms over ℚp, we ask how many variables are required to guarantee that the system has a nontrivial zero. We show that if the prime p satisfies p > (largest degree) − (smallest degree) + 1, then there is a bound on the sufficient number of variables which is a polynomial in the degrees of the forms.

On systems of diagonal forms

In this paper we consider systems of diagonal forms with integer coefficients in which each form has a different degree. Every such system has a nontrivial zero in every p-adic field Qp provided that the number of variables is sufficiently large in terms of the degrees. While the number of variables required grows at least exponentially as the degrees and number of forms increase, it is known that if p is sufficiently large then only a small polynomial bound is required to ensure zeros in Qp. In this paper we explore the question of how small we can make the prime p and still have a polynomial bound. In particular, we show that we may allow p to be smaller than the largest of the degrees.

A CHARACTERIZATION OF NONSINGULAR PAIRS OF QUADRATIC FORMS

Let F, G be a pair of quadratic forms defined over an arbitrary field k. We give a characterization for when every nontrivial zero of F = G = 0 defined over the algebraic closure of k is nonsingular. When chark ≠ 2, this result is well known. When chark = 2, the problem divides into two cases. If n is odd, we use the half-determinant, and if n is even, we use the Arf invariant for this characterization. The characterization depends only on the coefficients of the quadratic forms and operations taking place in the field k.

Normal characterizations of lattices

LetXbe an arbitrary nonempty set andℒa lattice of subsets ofXsuch that∅,X∈ℒ. Let𝒜(ℒ)denote the algebra generated byℒandI(ℒ)denote those nontrivial, zero-one valued, finitely additive measures on𝒜(ℒ). In this paper, we discuss some of the normal characterizations of lattices in terms of the associated lattice regular measures, filters and outer measures. We consider the interplay between normal lattices, regularity orσ-smoothness properties of measures, lattice topological properties and filter correspondence. Finally, we start a study of slightly, mildly and strongly normal lattices and express then some of these results in terms of the generalized Wallman spaces.