Jump sets in local fields

We show how to use the combinatorial notion of jump sets to parametrize the possible structures of the group of principal units of local fields, viewed as filtered modules. We establish a natural bijection between the set of jump sets and the orbit space of a $p$-adic group of filtered automorphisms acting on a free filtered module. This, together with a Markov process on Eisenstein polynomials, culminates into a mass-formula for unit filtrations. As a bonus the proof leads in many cases to explicit invariants of Eisenstein polynomials, yielding a link between the filtered structure of the unit group and ramification theory. Finally, with the basic theory of filtered modules developed here, we recover, with a more conceptual proof, a classification, due to Miki, of the possible sets of upper jumps of a wild character: these are all jump sets, with a set of exceptions explicitly prescribed by the jump set of the local field and the size of its residue field

Improve Your Legal Writing

Christine Pagano of Golden Gate University School of Law suggests some helpful resources for attorneys wishing to hone their drafting skills

The Issue of Unemployment Among People with Disabilities

The rate of unemployment for people with disabilities continues to rise greatly above that of people without disabilities. The issue seems to be exacerbated by employer biases and concerns which are not supported in the face of evidence. A lack of employer education on disability related subjects causes this misconception among both employers and the public as a whole. To resolve the underlying problem of miseducation, an increase in the self-identification of people with disabilities is necessary to provide researchers with data to assist in the formation of a revised curriculum

Improve Your Legal Writing

Christine Pagano of Golden Gate University School of Law suggests some helpful resources for attorneys wishing to hone their drafting skills

Unit equations and Fermat surfaces in positive characteristic

In this article we study the three-variable unit equation $x + y + z = 1$ to be solved in $x, y, z \in \mathcal{O}_S^\ast$, where $\mathcal{O}_S^\ast$ is the $S$-unit group of some global function field. We give upper bounds for the height of solutions and the number of solutions. We also apply these techniques to study the Fermat surface $x^N + y^N + z^N = 1$