Completeness of zero curve tracing for analytic functions

AbstractEmpirical data for the polynomial confirm the efficiency of exploring function structure as a means of isolating the zeros of a scalar analytic function defined on a disk (Klip, 1985). In exceptional cases the tracing is not complete which means that certain arcs may not have been traced. A mathematical basis for the algorithm which investigates and restores completeness, the so-called completeness algorithm, is presented. The main theorem is that completeness of the tracing, as verified at the isolated zeros, is sufficient for completeness of the tracing. Its proof evolves along various lemmas, one of which provides a new condition for the location of the critical points. The paper concludes with a brief description of the completeness algorithm