This project undertakes the implementation of an algebraic-geometric toolkit, written in C, which can solve systems of equations over the complex numbers, and inequalities over the reals. There are many applications of such a toolkit, since problems from many branches of science and engineering can be formulated using systems of polynomial equations and inequalities. By several well-known theorems, in particular, Tarski's theorem for the reals, it is possible to solve such problems in principle. But the worst case bounds are exponential or doubly-exponential in the number of variables, and no practical system has appeared that can deal with more than a few special cases. On the other hand, systems of polynomials may have a special structure, a kind of sparseness, that implies a complexity (measured by the algebraic degree) much lower than the worst case. Indeed, most of the important applications are of this type. Very recently, algorithms have been developed that can exploit this structure. This realization is a strong motivation for developing the toolkit at this time. In addition, there have been improvements in algorithms for sign determination and symbolic-numeric computation, and it is felt that these methods are now advanced enough to warrant implementation.
