On the Pierce-Birkhoff Conjecture

This paper represents a step in our program towards the proof of the Pierce--Birkhoff conjecture. In the nineteen eighties J. Madden proved that the Pierce-Birkhoff conjecture for a ring A$is equivalent to a statement about an arbitrary pair of points$\alpha,\beta\in\sper\ A$and their separating ideal$$; we refer to this statement as the Local Pierce-Birkhoff conjecture at$\alpha,\beta$. In this paper, for each pair$(\alpha,\beta)$with$ht()=\dim A$, we define a natural number, called complexity of$(\alpha,\beta)$. Complexity 0 corresponds to the case when one of the points$\alpha,\beta$is monomial; this case was already settled in all dimensions in a preceding paper. Here we introduce a new conjecture, called the Strong Connectedness conjecture, and prove that the strong connectedness conjecture in dimension n-1 implies the connectedness conjecture in dimension n in the case when$ht()$is less than n-1. We prove the Strong Connectedness conjecture in dimension 2, which gives the Connectedness and the Pierce--Birkhoff conjectures in any dimension in the case when$ht()$less than 2. Finally, we prove the Connectedness (and hence also the Pierce--Birkhoff) conjecture in the case when dimension of A is equal to$ht()=3$, the pair$(\alpha,\beta)$is of complexity 1 and$A$ is excellent with residue field the field of real numbers

Franklin Pierce Law Center: Leading the Way in Legal Education for New Hampshire

[Excerpt] This issue of the Pierce Law Review is the first devoted entirely to the practice of law in New Hampshire. This venture is appropriate because the Franklin Pierce Law Center is the only law school in the State. We are truly New Hampshire’s law school. Our Trustees, faculty, staff, and students feel this responsibility profoundly. Pierce Law serves as both a state law school and a national and international school. While we send a greater percentage of our graduates out of state than any other law school in the country except one, our alumni comprise fully one-third of the lawyers in New Hampshire. (I should point out, too, that we have almost one thousand alumni in over one hundred countries.

Approximation of quadratic irrationals and their pierce expansions

In this article two aims are pursued: on the one hand, to present a rapidly converging algorithm for the approximation of square roots; on the other hand and based on the previous algorithm, to find the Pierce expansions of a certain class of quadratic irrationals as an alternative way to the method presented in 1984 by J.O. Shallit; we extend the method to find also the Pierce expansions of quadratic irrationals of the form $2 (p-1) (p - \sqrt{p^2 - 1})$ which are not covered in Shallit's work.Quadratic irrationals, Pierce series

Approximate roots of a valuation and the Pierce-Birkhoff Conjecture

This paper is a step in our program for proving the Piece-Birkhoff Conjecture for regular rings of any dimension (this would contain, in particular, the classical Pierce-Birkhoff conjecture which deals with polynomial rings over a real closed field). We first recall the Connectedness and the Definable Connectedness conjectures, both of which imply the Pierce - Birkhoff conjecture. Then we introduce the notion of a system of approximate roots of a valuation v on a ring A (that is, a collection Q of elements of A such that every v-ideal is generated by products of elements of Q). We use approximate roots to give explicit formulae for sets in the real spectrum of A which we strongly believe to satisfy the conclusion of the Definable Connectedness conjecture. We prove this claim in the special case of dimension 2. This proves the Pierce-Birkhoff conjecture for arbitrary regular 2-dimensional rings

Professor Bryan Harris Remembered: Volez to a Pierce Law Friend

Bryan Harris, MA (Oxon), passed away recently in his beloved native England, after a brief illness. His wife Mary, two sons and a daughter survive him. Bryan Harris had a long and distinguished career as an author, educator, barrister, diplomat, publisher and lobbyist. He was a consultant on European Union policies and laws to commercial and professional firms and associations. For almost three decades he was a Member of the Board of Trustees and Adjunct Professor of European Union Law at Pierce Law. Pierce Law President and Dean, John Hutson summed up what many members of the Pierce Law community expressed to me as I prepared this tribute saying, I think of Bryan mostly in single words ... jovial, cheerful, humble, dignified, diplomatic, caring ... Dean Huston shared that Professor Harris will be recognized during the 2004 Commencement

On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields

We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if W is such a variety, then every piecewise polynomial function on W can be written as suprema of infima of polynomial functions on W. More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of such varieties, which implies the Pierce-Birkhoff Conjecture.Comment: v2: Removed typos, changed content. v3: Added missing conditions for several results in section