Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation ( Abstract ) ∗

Methods for computing triangular decompositions of polynomial systems can be classified into two groups. First, those computing a series of regular chains C1 . , Ce such that for each irreducible component V of the variety of the input system, one of the Ci’s encodes a generic zero of V . An example is the algorithm of Michael Kalkbrener in his PhD thesis. Secondly are those methods computing a series of characteristic sets C1 . , Cf (in the sense of Wu Wen Tsün) such that the variety of the input system is the union of the quasi-components of the Ci’s. A large variety of methods fall in the second family, in particular the one proposed in 1987 by Wu Wen Tsün in the 1st volume of the MM Research Preprints. Some methods belong to both families, this is the case for those proceeding in an incremental manner, that is, solving one equation after another. These latter methods rely on an operation for computing the intersection of an hypersurface and the quasi-component of a regular chain. This is an attractive operation since its input can be regarded as well-behaved geometrical objects. However, known algorithms (the one of Daniel Lazard in 1991 and the one of the second author in 2000) are quite involved and difficult to analyze. We revisit this intersection operation. We show that under “genericity assumptions” a simple algorithm can be stated and analyzed. In this context, it follows a classical projection-extension scheme. However, we shall see that the cost of the extension step can be neglected in comparison to that of the projection, which itself “essentially” reduces to a cost that all such intersection algorithms have to pay. In our experimental results, realized with the RegularChains library in Maple, our new intersection outperforms the one of the second author by several orders of magnitude on sufficiently difficult problems. Root Isolation of Zero-dimensional Polynomial Systems with Linear Univariate Representation(Abstract) ∗ Jin-San Cheng, Xiao-Shan Gao, Leilei Guo Key Lab of Mathematics Mechanization Institute of Systems Science, Academia Sinica, Beijing, China xgao@mmrc.iss. ac.cn,jcheng@amss.ac.cn In this paper, we present a new explicit representation for the roots of a zero-dimensional polynomial system: the roots are represented as linear combinations of some roots of several univariate polynomial equations, which is a generalization of the method proposed in [1]. We call this representation as Linear Univariate Representation (shortly LUR). This gives an algebraic representation for the roots of the system. It is different from the wellknown RUR representation which is based on the idea of generic position [3]. From LUR, we can obtain the roots of the system under any given precision by refining the isolation intervals of these univariate polynomial equations at most once. 1 Linear univariate representation Let P = {f1(x1, . . . , xn), . , fs(x1, . . . , xn)} be a zero-dimensional polynomial system in Q[x1, . , xn], where Q is the field of rational numbers. We use VC(P) to denote its roots in C, where C is the field of complex numbers. Denote (P) to be the ideal generated by P and Ii = (P) ∩ Q[x1, . , xi], i = 1, . , n. By a linear univariate representation (abbr. LUR), we mean a set like {T1(x), . , Tn(x), si, di, i = 1, . , n − 1} (1) where Ti ∈ Q[x] are univariate polynomials, si and di are positive rational numbers. The roots of (1) are defined to be {(α1, α2 − α1 s1 , . . . , αn − αn−1 s1 · · · sn−1 ) |Ti(αi) = 0, i = 1, . , n and |αi+1 − αi| < s0 · · · si−1di, i = 1, . , n − 1, where s0 = 1. } (2) An LUR for P = 0 is a set of form (1) whose roots are exactly the roots of P = 0. ∗Partially supported by a National Key Basic Research Project of China and by a USA NSF grant CCR-0201253.