Bounds for the vanishing order for solutions of a linear differential system
The author provides upper bounds for the order of vanishing of a polynomial restricted to a solution of a linear differential system through a nonsingular point. He proves the following main result. Let \(L\) be a linear differential polynomial of order \(n\) over \(C(x)\), \(a_1,..., a_{s}\) be singular points of \(L\) in \(CP^{1}\), and \(m_{L}=\sum_{i=1}^{s}m_{i}\), where \(m_{i}\) is the maximum multiplicity of the pole \(a_{i}\) in the coefficients of \(L\). Also let \(u\) be a zero of \(L\), \(C{u}\) be a differential ring and \(V\) be a differential vector space generated by \(u\) over \(C, R\) be a ring of convergent power series over \(C, \omega\) be the usual order function on \(R\), and \(\varphi:C{u}\longrightarrow R\) be a differential inclusion over \(C\). Then, for any \(v\in V\), \(\omega(\varphi(v))\leq (n-1) + \frac{n(n-1)}{2}(m_{L}-2)\).
scientific article; zbMATH DE number 1930207