On the linear independence of the values of \(E\)-functions at algebraic points (Q1898279)

Let (1) \(f_1 (z), \dots, f_n (z)\) be the set of \(E\)-functions satisfying the system of differential equations \(y_k'= \sum_{i=1}^n Q_{ki} y_i\), \(k= 1,\dots, n\), \(Q_{k,i}\in \mathbb{C}(z)\), let \(T(z)\in \mathbb{C}[z]\) be the least common denominator of all \(Q_{k,i}\). Denote \(A= \overline {\mathbb{Q}}\), \(\mathbb{K}= \mathbb{Q} (\theta)\) which is an algebraic field over \(\mathbb{Q}\) containing all the coefficients of \(E\)-functions (1) and the numbers \(\xi\) \((\xi\in A)\), and \(h=[\mathbb{K}: \mathbb{Q}]\), \(h\geq 1\). In this paper, the author proposes three hypotheses: H.(A): If the functions (1) are linearly independent over \(\mathbb{C} (z)\), the numbers (2) \(f_1 (\xi), \dots, f_n (\xi)\) are linearly independent over \(A\). H.(B): Under the assumptions of H.(A), the numbers (2) are linearly independent over \(\mathbb{K}\). H.(C): If the functions (1) are not connected by a homogeneous algebraic equation over \(\mathbb{C} (z)\) the numbers (2) are linearly independent over \(\mathbb{K}\). Partial results have been proved for these hypotheses in this paper.