Algebraic complexity theory

Algebraic complexity theory investigates the computational cost of solving problems with an algebraic flavor. Several cost measures are of interest. We consider arithmetic circuits, which can perform the (exact) arithmetic operations +, -, *, / at unit cost, and take their size ( = sequential time) or their depth (=parallel time) as cost functions. This is a natural "structured" model of computation for the computation of rational func­ tions over any ground field. If inputs can be represented by strings over a finite alphabet-as is the case for polynomials over OJ-we can also use a "general" model such as Turing machines or Boolean circuits. The complexity of a problem is the minimal cost (in the measure under con­ sideration) sufficient to solve it. Its investigation splits into two tasks, which require very different methodologies. The first task is the design of good algorithms, proving upper bounds on the complexity. The second, usually more difficult task, is the discovery of intrinsic properties ("in­ variants") of problems, and estimation of the progress that an algorithm can make, say step by step, in terms of these invariants, thus proving lower bounds on the cost of any conceivable algorithm. Within the wider field of complexity theory, few areas have had similar success in establishing matching upper and lower bounds on the complexity of many natural problems. Our subject takes its questions from computer science, mainly numerical and symbolic computation. The approach is mathematical, and some problems, by their nature, require fairly sophisticated methods. Classifying our problems under the perspective of polynomial time, they fall into three categories. In the first category (Sections 2 and 3