Integer Points in Backward Orbits

A theorem of J. Silverman states that a forward orbit of a rational map $\phi(z)$ on $\mathbb P^1(K)$ contains finitely many $S$-integers in the number field $K$ when $(\phi\circ\phi)(z)$ is not a polynomial. We state an analogous conjecture for the backward orbits using a general $S$-integrality notion based on the Galois conjugates of points. This conjecture is proven for the map $\phi(z)=z^d$, and consequently Chebyshev polynomials, by uniformly bounding the number of Galois orbits for $z^n-\beta$ when $\beta\not =0$ is a non-root of unity. In general, our conjecture is true provided that the number of Galois orbits for $\phi^n(z)-\beta$ is bounded independently of $n$.Comment: 13 page

Towards Evaluating the Quality of a Spreadsheet: The Case of the Analytical Spreadsheet Model

We consider the challenge of creating guidelines to evaluate the quality of a spreadsheet model. We suggest four principles. First, state the domain-the spreadsheets to which the guidelines apply. Second, distinguish between the process by which a spreadsheet is constructed from the resulting spreadsheet artifact. Third, guidelines should be written in terms of the artifact, independent of the process. Fourth, the meaning of "quality" must be defined. We illustrate these principles with an example. We define the domain of "analytical spreadsheet models", which are used in business, finance, engineering, and science. We propose for discussion a framework and terminology for evaluating the quality of analytical spreadsheet models. This framework categorizes and generalizes the findings of previous work on the more narrow domain of financial spreadsheet models. We suggest that the ultimate goal is a set of guidelines for an evaluator, and a checklist for a developer.Comment: Proc. European Spreadsheet Risks Int. Grp. (EuSpRIG) 2011 ISBN 978-0-9566256-9-