Estimates for polynomials and linear forms in direct products of fields
type: Item
instance of: scholarly article
title: Estimates for polynomials and linear forms in direct products of fields
zbMATH Open document ID: 0879.11041
published in: Moscow University Mathematics Bulletin
publication date: 1998-01-18T00:00:00Z
review text: 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/ \sqr…
Mathematics Subject Classification ID: 11J91
zbMATH DE Number: 1028442
zbMATH Keywords: direct product of fields
zbMATH Keywords: estimates of values of polynomials
zbMATH Keywords: linear forms
zbMATH Keywords: global relations
author: V. G. Chirskii
reviewed by: Yaochen Zhu
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1028442