Estimates for polynomials and linear forms in direct products of fields
Let \(K\) be a number field. For every place \(v\) of \(K\), let \(K_v\) be the corresponding completion of \(K\). If \(a\) is an element of the direct product of a certain collection of fields \(K_v\), and the set of values of its normalized coordinates is bounded, then we denote the upper bound of this set by \(|a|\). For a power series \(f(z)= \sum^\infty_{\nu=0} a_\nu \nu! z^\nu\), \(a_\nu \in K\), and a point \(\xi\in K\), let \(f(\xi)\) be the corresponding element of the direct product of \(K_v\), in which the series \(f(\xi)\) converges. We say that the series \(f(z)\) belongs to the class \(F(K,c_1,c_2,c_3,q_0)\) if (i) \(a_\nu\in K\) and \(\house{a_\nu}\leq \exp (c_1\nu)\) for \(\nu=0,1, \dots\), where \(\house{\alpha}\) denotes the largest of the moduli of the conjugates of \(\alpha\in K\); and (ii) there is a sequence \(d_n=q^n_0d_{0,n}\) with \(q_0\), \(d_{0,n} \in\mathbb N\) such that \(d_na_\nu \in \mathbb Z_K\) for \(n=0,1, \dots\) and \(\nu= 0, 1, \dots,n\), and \(d_{0,n}\) are only divisible by primes \(p<c_2n\) and \(\text{ord}_p d_{0,n} \leq c_3 \log_pn\). By use of the Siegel-Shidlovskii method the author proves two theorems about estimates of values of polynomials and linear forms for the class \(F(K,c_1, c_2,c_3, q_0)\). One of them is as follows: Let the series \(f_1(z), \dots, f_m(z)\) belong to \(F(K,c_1, c_2, c_3, q_0)\), form the solution of the system of equations \(y_i'= \sum^m_{i=1} A_{i,j}y_j\), \(i=1,2, \dots, m\), where \(A_{i,j} \in K(z)\), and be algebraically independent over \(K(z)\). Let \(\xi= a/q\), \(a\in \mathbb Z_K\), \(q\in \mathbb N\) with \(\xi T(\xi) \neq 0\) (here \(T\in \mathbb Z_K[z]\) with \(TA_{i,j} \in \mathbb Z_K [z])\). Suppose that \(P(x_1, \dots, x_m) \in\mathbb Z [x_1, \dots, x_m]\) is any nonzero polynomial of total degree \(d\) and of height \(H(P) \leq H\) with \(H\geq H_0\). Then the inequality \[ \biggl|P\bigl( {\mathbf f}_1 (\xi), \dots, {\mathbf f}_m (\xi)\bigr) \biggr|>H^{-{m+ d \choose m} -c_4/ \sqrt {\ln\ln H}} \] holds, where \({\mathbf f}_1 (\xi), \dots, {\mathbf f}_m(\xi)\) are the elements of the direct product of the fields \(K_v\) for which \(v\mid p\), \((p,qq_0)=1\) and \[ p\leq{m+d \choose m} {\ln H \over \ln\ln H} (1+c_5/ \sqrt {\ln\ln H}). \] These results are quantitative generalizations of the theorems concerning the global relations for \(F(K,c_1, c_2,c_3, q_0)\) given by the author [Math. Notes 48, No. 2, 795-798 (1990); translation from Mat. Zametki 48, No. 2, 123--127 (1990; Zbl 0764.11031)].
scientific article; zbMATH DE number 1028442