Computational Complexity in Polynomial Algebra

In recent years a number of algorithms have been designed for the "inverse" computational problems of polynomial algebra—factoring polynomials, solving systems of polynomial equations, or systems of polynomial inequalities, and related problems—with running time considerably less than that of the algorithms which were previously known. (For the computational complexity of the "direct" problems such as polynomial multiplication or determination of g.c.d.'s see [1, 16] and also [9].) It should be remarked that as a result a hierarchical relationship between the computational problems of polynomial algebra, from the point of view of computational complexity, has been elucidated. The successful design of these algorithms depended to a large degree on developing them in the correct order: first the algorithms for the problems which are easier in the sense of this hierarchy were designed, which were then applied as subroutines in the solutions of more difficult problems. So far problems of the type discussed here have been considered easier only when they are special cases of the more difficult ones; e.g., the solution of a system of polynomial equations is considered as a particular case of quantifier elimination. A powerful impetus for this development came initially from the development of polynomial-time algorithms for factoring polynomials. On the other hand, a major role has been played by a new insight from the computational point of view: treating the solution of systems of polynomial equations in the framework of the determination of the irreducible components of an algebraic variety. This has made it possible to apply the polynomial factorization algorithm to this problem. In addition a successful reduction of the problem of solving systems of polynomial inequalities to the "nonspecial" case of this problem was achieved by means of an explicit use of infinitesimals in the calculations, and the "nonspecial" case was in turn reduced to the solution of a suitable system of polynomial equations. Finally, for the design of decision procedures for the first order theories of algebraically closed or real closed fields, appropriate solvability criteria for the corresponding systems with variable coefficients were produced which are "uniform" in the set of auxiliary parameters.