The problem of large powers and that of large roots (Q2710600)

Let \(K\) be a field of the characteristic 0. There exist problems that can be solved by a finite number of operations. For example, if \(a_0,\ldots,a_n\) and \(b\) are elements of \(K,\) it is possible to find out if \(b\) is a root of a polynomial \(a_0+a_1x+\ldots+a_nx^n,\) using Horner's algorithm, and a test for the equality to zero. Such calculations can be presented by special graphs. The natural question is: what is the minimal number of operations sufficient for solving a concrete problem? Let's enrich the given field with differentiation (and so deal with a differential field). Is it true that the minimal number of operations sufficient for solving a concrete problem will be reduced? The author investigates two problems, the problem of large powers and the problem of large roots, for which differentiation do not accelerate calculations. The author uses calculation schemes from \textit{B. Poizat} [Les petits cailloux. Lyon: Aléas (1995; Zbl 0832.68044)].