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 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 "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 and complexity bounds (we shall state two of them).It should be well understood that our method does not improve the well known 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].

The Computational Complexity of the Chow Form

“…To define this parameter, which is a suitable generalization of the geometric degree of a 0-dimensional system introduced in [24], we first give the following definition: 1.…”

Bit complexity for multi-homogeneous polynomial system solving—Application to polynomial minimization

“…This data structure has a long history, going back to work of Kronecker and Macaulay [31,34], and has been used in a host of algorithms in effective algebra [17,19,1,20,18,40,21,33].…”

“…Following references such as [19,20,18,21,33], we will represent the input polynomials f of our algorithm by means of a straight-line program, that is, a sequence of elementary operations +, −, × that evaluates the polynomials f from the input variables X 1 , . .…”

Coevolutionary Architectures With Straight Line Programs for Solving the Symbolic Regression Problem

“…A particular class of straight line programs, known in the literature as arithmetic circuits, constitutes the underling computation model in Algebraic Complexity Theory (Burguisser et al, 1997). They have been used in linear algebra problems (Berkowitz, 1984), in quantifier elimination (Heintz et al, 1990) and in algebraic geometry (Giusti et al, 1997). Recently, slp's have been presented as a promising alternative to the trees in the field of Genetic Programming, with a good performance in solving some regression problem instances (Alonso et al, 2008).…”