Integral points of projective spaces omitting hyperplanes over function fields of positive characteristic
Let \(K\) be the function field of an irreducible smooth projective curve \(C\) of genus \(g\) defined over an algebraically closed field \(k\) of characteristic \(p>0\). The author proves some results concerning \(S\)-integral points in \(\mathbb{P}^n(K)\) minus \(2n+2\) hyperplanes. The precise statement is given below. The height of \((f_0,\ldots,f_n)\in \mathbb{P}^n(K)\) is given by \[ h(f_0,\ldots ,f_n)=-\sum_{P\in C(k)} \min ( v_P(f_0),\ldots ,v_P(f_n)), \] where \(v_P\) is the normalized valuation at \(P\). The height of \(a\in K\) is defined by \(h(a)=-\sum_{P\in C(k)}\min (0,v_P(a))\). Define \(2n+2\) linear forms \(L_0,\ldots ,L_{2n+1}\) in \(n+1\) variables with coefficients in \(K\) as follows: \(L_i =X_i\) for \(i=0,\ldots ,n\), and \(L_{n+1+i}=\sum_{i=0}^n a_{ij}X_j\) with \(a_{ij}\in K\). Suppose that \(L_0,\ldots ,L_n\) are in general position. Further suppose there is a positive integer \(m\) such that (*) the numbers \(\prod_{i=0}^n a_{i,\sigma (i)}\), where \(\sigma\) runs through the permutations of \(0,\ldots, n\), are linearly independent over \(K^{p^m}\). Let \(\widetilde{S}\) be the set of \(P\in C(k)\) such that \(v_P(\prod_{0\leq i,j\leq n} a_{ij})\not= 0\). Then for all \(S\)-integral points \((f_0,\ldots ,f_n)\in \mathbb{P}^n(K)\backslash \bigcup_{i=0}^{2n+1}\{ L_i=0\}\) one has \[ h(f_0,\ldots ,f_n)\leq {n(n+1)\over 2}p^{m-1}\max ( 0, 2g-2+|S\cup \widetilde {S}|) +\sum_{0\leq i,j\leq n} h(a_{ij}) . \] As a consequence the author shows that the number of \(S\)-integral points (up to constant factors in each coordinate) in \(\mathbb{P}^n(K)\backslash \cup_{i=0}^{2n+1}\{ L_i=0\}\) is at most \[ \Big( n(n+1)p^{m-1}\max (0,2g-2+|S\cup \widetilde{S}|)+1\Big)^{(n+1)|S|} . \] The author gives an example showing that the technical condition (*) cannot be removed completely. The author's main tool is her \(n\)-dimensional generalization [J. Number Theory 58, 139-157 (1996; Zbl 0866.11066)] of \textit{R. C. Mason}'s abc-theorem over function fields of positive characteristic [LMS Lecture Notes 96, Cambridge Univ. Press (1984; Zbl 0533.10012), Chap. VI].
Diophantine equations over function fields
scientific article; zbMATH DE number 1375115