Straight-line programs in Geometric Elimination Theory

We present a new method for solving symbolically zero-dimensional polynomial equation systems in the affine and toric case. The main feature of our method is the use of problem adapted data structures: arithmetic networks and straight--line programs. For sequential time complexity measured by network size we obtain the following result: it is possible to solve any affine or toric zero-dimensional equation system in non--uniform sequential time which is polynomial in the length of the input description and the "geometric degree " of the equation system. Here, the input is thought to be given by a straight-line program (or alternatively in sparse representation), and the length of the input is measured by number of variables, degree of equations and size of the program (or sparsity of the equations). The geometric degree of the input system has to be adequately defined. It is always bounded by the algebraic-combinatoric "B'ezout number " of the system which is given by the Hilbert function of a suitable homogeneous ideal. However, in many important cases, the value of the geometri