Algorithms in Semi-algabraic Geometry
1:45 p.m., Wednesday, July 17, 1996
Hooker Auditorium, Willets-Hallowell Center
Mt. Holyoke College
South Hadley, Massachusetts
The first part of the thesis deals mainly with the decision problem for the first order theory of real closed fields, and the more general problem of quantifier elimination. We give algorithms which improve the complexity of of all the previously known algorithms for these problems. Moreover, our techniques allow us to prove some purely mathematical theorems on the number of connected components and on the existence of small rational points in a given semi-algebraic set.
The second part of this work deals with connectivity questions of semi-algebraic sets. We develop new techniques in order to give an algorithm for computing roadmaps of semi-algebraic sets which improves on the complexity of the previous algorithms for this problem.
The third part of this work deals with bounding the topological complexity of semi-algebraic sets in terms of the number and the degrees of the polynomials describing it. We extend and improve a classical and widely used result of Oleinik and Petrovsky(1949), Thom (1965) and Milnor(1964), bounding the sum of the Betti numbers of semi-algebraic sets. Using the ideas behind this result, we give the first singly exponential algorithm for computing the Euler characteristic of an arbitrary semi-algebraic set.
One common thread that links these results is that our bounds are separated into a combinatorial part (the part depending on the number of polynomials) and an algebraic part (the part depending on the degrees of the polynomials). The combinatorial part of the complexity of our algorithms is frequently tight and this marks the improvement of many of our results. This is most striking when one considers that in many applications, for instance in computational geometry, it is the number of polynomials which is the most important parameter (the degrees and the number of variables are usually small). Another important and new feature of some of our results is that when the given semi-algebraic set is contained in a lower dimensional variety, the combinatorial part of the complexity depends on the dimension of this variety rather than on the dimension of the ambient space. This is useful when one considers semi-algebraic sets which have low real dimension embedded in a higher dimensional space.