A new algebraic method for robot motion planning and real geometry

We present an algorithm which solves the findpath or generalized movers\´ problem in single exponential sequential time. This is the first algorithm for the problem whose sequential time bound is less than double exponential. In fact, the combinatorial exponent of the algorithm is equal to the number of degrees of freedom, making it worst-case optimal, and equaling or improving the time bounds of many special purpose algorithms. The algorithm accepts a formula for a semi-algebraic set S describing the set of free configurations and produces a one-dimensional skeleton or "roadmap" of the set, which is connected within each connected component of S. Additional points may be linked to the roadmap in linear time. Our method draws from results of singularity theory, and in particular makes use of the notion of stratified sets as an efficient alternative to cell decomposition. We introduce an algebraic tool called the multivariate resultant which gives a necessary and sufficient condition for a system of homogeneous polynomials to have a solution, and show that it can be computed in polynomial parallel time. Among the consequences of this result are new methods for quantifier elimination and an improved gap theorem for the absolute value of roots of a system of polynomials.