Effective estimates of the measure of algebraic independence of the values of E-functions (Q1112100)

The author gives an elegant proof of the following explicit version of one of his previous fundamental results: Let every non-zero solution \(F(z)=(f_ 0(z),...,f_ m(z))\) of the system of differential equations \[ (1)\quad y_ k'=\sum^{m}_{i=0}q_{ki}y_ i,\quad k=0,1,...,m,\quad q_{ki}\in {\mathbb{C}}(z) \] be homogeneously algebraically independent over \({\mathbb{C}}(z)\). Let the corresponding differential operator \(D=\partial /\partial z+\sum_{k}(\sum_{i}q_{ki}x_ i)\partial /\partial x_ k\) act on \({\mathbb{C}}(z)[x_ 0,...,x_ m]\). Let the non-zero radical ideal \(J\subset {\mathbb{C}}[z,x_ 0,...,x_ m]\) be homogeneous with respect to \(x_ 0,...,x_ m\) and satisfy t(z)DJ\(\subset J\), where t(z) is the least common denominator of the \(q_{ki}\). Then \(t(z)x_ 0... x_ m\in J\), and consequently for every non-zero solution F(z) of (1) holomorphic at the origin, \(\min_{P\in J}ord P(z,F(z))\leq C,\) where ord denotes the order of zero at the origin and \(C=ord t(z)f_ 0(z)... f_ m(z).\) Previously the author had shown the existence of an unspecified finite bound C under the weaker hypothesis that the particular solution F(z) be homogeneously algebraically independent over \({\mathbb{C}}(z)\). The reviewer had determined C when every non-zero solution is algebraically independent over \({\mathbb{C}}(z)\). The author deduces effective measures of algebraic independence of values of E-function solutions of related inhomogeneous differential equations at algebraic non-singular points.