Linear independence of values of \(E\)-functions related by a single algebraic equation over the field of rational functions (Q1922312)

Let \(f_1(z), \dots, f_m(z)\) be \(KE\)-functions satisfying a system of linear homogeneous differential equations and \(A\) denote the field of all algebraic numbers. Assume \(f_1(z), \dots, f_m(z)\) are linearly independent over \(\mathbb{C}(z)\) of homogeneous transcendence degree \(m-1\). Further let \[ P(z,x_1, \dots, x_m)\in K[z] [x_1, \dots, x_m] \] be a homogeneous irreducible primitive polynomial such that \(P(z,f_1(z), \dots, f_m(z))=0\). The author proves that under the above conditions the numbers \(f_1(\xi), \dots, f_m(\xi)\) \((\xi\neq 0\), \(\xi\in A)\) are linearly dependent over \(A\) if and only if the decomposition into irreducible factors over \(A\) of the polynomial \(P(\xi,x_1, \dots, x_m)\) contains a linear form \(L=L(x_1, \dots, x_m)\in A[x_1, \dots, x_m]\), \(L\not \equiv 0\), such that \(L(f_1(\xi), \dots, f_m(\xi))=0\). This implies that if \(P(\xi,x_1, \dots, x_m)\) is irreducible then \(f_1(\xi), \dots, f_m(\xi)\) are linearly independent over \(A\). The author also gives an analogue in the nonhomogeneous case and the measure of linear independence in the numbers \(f_1(\xi), \dots, f_m(\xi)\).