In 1982-1983, E. Nochka proved a conjecture of Cartan on defects of holomorphic curves in P-n relative to a possibly degenerate set of hyperplanes. This was further explained by W. Chen in his 1987 thesis, and subsequently simplified by M. Ru and P.-M. Wong in 1991. The proof involved assigning weights to the hyperplanes. This paper provides further simplification of the proof of the construction of the weights, by bringing back the use of the convex hull in working with the "Nochka diagram." (c) 2006 Elsevier Inc. All rights reserved.

Let $K$ be a number field or the function field of a curve over an algebraically closed field of characteristic 0. Let $n\geq 2$, and let $f(X)\in K[X]$ be a polynomial of degree $d\geq 2$. We present two

arithmetic properties of the dynamics of the coordinate-wise self-map $\varphi=f\times\ldots\times f$ of $(\bP^1)^n$, namely the dynamical analogs of the Hasse principle and the Bombieri-Masser-Zannier height bound theorem. In particular, we prove that the Hasse principle holds when we intersect an orbit and a preperiodic subvariety, and that the intersection of a curve with the union of all periodic hypersurfaces have bounded heights unless that curve is vertical or contained in a periodic hypersurface. A common crucial ingredient for the proof of these two properties is a recent classification of $\varphi$-periodic subvarieties by Medvedev-Scanlon. We also present the problem of primitive prime divisors in dynamical sequences by Ingram-Silverman which is needed and closely related to the dynamical Hasse principle. Further questions on the bounded height result, and a possible generalization of the Medvedev-Scanlon classification are briefly given at the end.