Subspace theorems for homogeneous polynomial forms (Q877465)

The authors prove among other things the following extension of the Subspace Theorem. Let \(K\) be an algebraic number field and let \(M_K\) denote its set of places. Choose a set of normalized absolute values \(\| \cdot\| _{\rho}\) \((\rho\in M_K )\) such that the product formula \(\prod_{\rho\in M_K}\| x \| _{\rho}=1 \) for \(x\in K^*\) holds. For vectors \(\xi =(x_0,\ldots ,x_n)\in K^{n+1}\), \(\rho\in M_K\), put \(\| \xi \| _{\rho}:=\max_i\| x_i\| _{\rho}\). Let \(S\) a finite set of places of \(K\) containing all archimedean places, and for \(\rho\in S\), let \(f_{\rho,i}\) \((i=0,\ldots, n)\) be homogeneous polynomials of degree \(m\geq 1\) in \(K[X_0,\dots, X_n]\) such that (*) \(\{ \xi\in\overline{K}^{n+1}: f_{\rho ,i}(\xi )=0\;(i=0,\dots ,n)\}=\{ 0\} \) for \(\rho\in S\). Then for every \(\varepsilon >0\) there exist a finite number of hypersurfaces \(Y_1,\ldots ,Y_s\) of \(\overline{K}^{n+1}\) such that for all \(S\)-integral points \(\xi\) in \(K^{n+1}\setminus\bigcup_i Y_i\) we have \[ \prod_{\rho\in S}\;\prod_{i=0}^n \| f_{\rho ,i}(\xi )\| _{\rho} \geq \Bigl(\max_{\rho\in S}\| \xi\| _{\rho}\Bigr)^{-\varepsilon}. \] This is a variation on a result proved earlier by \textit{P. Corvaja} and \textit{U. Zannier} [Am. J. Math. 126, 1033--1055 (2004; Zbl 1125.11022)]. Further, the authors' method of proof is similar to that of Corvaja and Zannier. The authors give a reformulation of their result which shows the analogy with Shifman's conjecture from value distribution theory, recently established by \textit{M. Ru} [Am. J. Math. 126, 215--226 (2004; Zbl 1044.32009)]. Further, the authors state some problems, among which to generalize to higher degree polynomials an extension of the Subspace Theorem due to \textit{M. Ru} and \textit{P.-M. Wong} [Invent. Math. 106, 195--216 (1991; Zbl 0758.14007)]. Reviewer's remark: Independently of the authors, Ferretti and the reviewer obtained a generalization of the authors' result in which the solutions \(\xi\) are taken from any \(n\)-dimensional algebraic subvariety \(X\) of \(K^{N+1}\), and condition (*) is replaced by \(\{ \xi\in X(\overline{K}): f_{\rho ,i}(\xi )=0\;(i=0,\ldots ,n)\}=\{ 0\}\) for \(\rho\in S\). Further, Ferretti and the reviewer obtained a quantitative version, with explicit upper bounds for the number \(s\) for the varieties \(Y_1,\ldots ,Y_s\) and for their degrees. The paper of Ferretti and the reviewer was submitted a couple of years ago to the Proceedings on Diophantine approximation in honour of the 70th birthday of W. Schmidt. It is expected that these proceedings are published in the course of 2008 by Springer Verlag.