On estimates of linear forms in values of E-functions (Q792375)

Continuing the author's earlier work on E-functions [Vestn. Mosk. Univ., Ser. I 1978, No.2, 3-12 (1978; Zbl 0386.10017)], the following results are announced. Let K be an algebraic number field, \({\mathcal O}_ K\) the domain of integers of K. Let \(\{f_{ij}(z)\}\), 1\(\leq i\leq m\), \(1\leq j\leq n_ i\) be a set of E-functions over K satisfying the system of differential equations \(y'\!_{ij}=Q_{ij0}(z)+\sum^{n_ i}_{s=0}Q_{ijs}(z) y_{is},\quad i=1,...,m,\quad j=1,...,n_ i.\) I. If \(K={\mathbb{Q}}\), \(\xi\neq 0\) lies in \({\mathbb{Q}}\) and is not a pole of any \(Q_{ijs}(z)\), then there exist constants \(c>0\) and \(a_ 0>0\), such that \[ \max_{i,j}(a_ i \| f_{ij}(\xi) x\|)>a^{-c/\sqrt{\ln \ln a}} \] for all natural numbers \(a_ 1,...,a_ m\) with \(a=\max_{i}a_ i\geq a_ 0\), where x is an integer with \(0<| x|<N^{-N}a_ 1^{n_ 1}...a_ m^{n_ m}\), \(N=n_ 1+...+n_ m.\) II. If K is of degree h over \({\mathbb{Q}}\), and \({\check \zeta}\in K\) satisfies the above hypothesis, then there exist constants \(c>0\) and \(a_ 0>0\) such that \[ \max_{1\leq t\leq h}| \ell_ t(\xi_ t)|>a^{-c/\sqrt{\ln \ln a}}\prod^{m}_{i=1}a_ i^{-n_ i} \] where \(a=\max_{i}a_ i\geq a_ 0\), \(a_ i=\max(1,\max_{1\leq j\leq n_ i}| \overline{a_{ij}}|)\); the linear forms \(\ell_ t(\xi_ t)\) are obtained from the linear form \(\ell(z)=b+\sum^{m}_{i=1}\sum^{n_ i}_{j=1}a_{ij}f_{ij}(z),\quad b,\quad a_{ij}\in {\mathcal O}_ K\) by replacing \(z={\check \zeta}\), \(a_{ij}\), b and all the coefficients of \(f_{ij}(z)\) by their conjugates. The reviewer [J. Aust. Math. Soc., Ser. A 35, 338-348 (1983; Zbl 0534.10027)] also proved results similar to the last ones, but the constants c and \(a_ 0\) can be expressed explicitly.