Straight-line programs in geometric elimination theory

Abstract

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 nonuniform 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 “Bézout 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 geometric degree of the system is much smaller than its Bézout number since this geometric degree does not take into account multiplicities or degrees of extraneous components (which may appear at infinity in the affine case or may be contained in some coordinate hyperplane in the toric case). Our method contains a new application of a classic tool to symbolic computation: we use Newton iteration in order to simplify straight-line programs occurring in elimination procedures. Our new technique allows for practical implementations, a meaningful characterization of the intrinsic algebraic complexity of typic elimination problems and reduces the still unanswered question of their intrinsic bit complexity to algorithmic arithmetics. However, our algorithms are not rational anymore as are the usual ones in elimination theory. They require some restricted computing with algebraic numbers. This is due to its numeric ingredients (Newton iteration). Nevertheless, at least in the case of polynomial equation systems depending on parameters, the practical advantage of our method with respect to more traditional ones in symbolic and numeric computation is clearly visible. Our approach produces immediately a series of division theorems (effective NullStellensätze) with new and more differentiated degree of complexity bounds (we shall state two of them). It should be well understood that our method does not improve the well-know worst-case complexity bounds for zero-dimensional equation solving in symbolic and numeric computing. Part of the results of this paper were announced in [25].