4 MARTIJN BAARTSE AND KLAUS MEER Figure 2. Six-bar Stephenson-1I mechanism kinematic param- eters such as lengths of linkages, sizes of angles etc. have to be determined in the dimensional synthesis step. Having chosen the kind of mechanism that is suitable to solve the problem (structural synthesis), in the dimensional synthesis step the unknown kinematic dimensions of the chosen mechanism have to be calculated. Mathematically, the problem leads to a polynomial system that has to be solved either over the real or the complex numbers depending on the formalization. Though both the number of variables and the degrees of the involved equations remain moderate, computing a complete catalogue of solutions in many cases already is demanding. Note that of course not all solutions of the resulting polynomial system are meaningful from an engineering point of view. A first complete dimensional synthesis for Stephenson mechanisms has been performed in [92], for a general introduction to solution algorithms for such kinematic problems see [100]. An important numerical technique to practically solve polynomial systems are homotopy methods. Here, the basic idea for solving F (x) = 0 is to start with another polynomial system G that in a certain sense has a similar structure as F . The idea then is to build a homotopy between G and F and follow the zeros of G numerically to those of F . A typical homotopy used is the linear one H(x, t) := (1 t) · G(x) + t · F (x), 0 t 1. In order to follow this approach the zeros of the starting system should be easily computable. Homotopy methods for solving polynomial systems are a rich source for many interesting and demanding questions in quite different areas. Their analysis has seen tremendous progress in the last 20 years and is outside the scope of this survey. There will be contributions in this volume by leading experts (which the authors of the present paper are not!) in the area, see the articles by C. Beltr´ an, G. Malajovich, and M. Shub. We just point to some of the deep results obtained and recommend both the other contributions in this volume and the cited literature as starting point for getting deeper into homotopy methods. Recent complexity analysis for homotopy methods was strongly influenced by a series of five papers in the 1990’ies starting with [94] and authored by M. Shub and S. Smale. A question that remained open at the end of this series and now commonly is addressed as Smale’s 17th
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