We present lifting techniques for triangular decompositions of zero-dimensional varieties, that extend the range of the previous methods. We discuss complexity aspects, and report on a preliminary implementation. Our theoretical results are comforted by these experiments.

The RegularChains library. This library provides functionalities for computing modulo regular chains based on the algorithms of [7]. The operation PolynomialRing allows the user to define the polynomial ring in which the computations take place and the order of the variables. The field of coefficients can be Q, a prime field, or a field of multivariate rational functions over Q or a prime field. Let us summarize the most frequently used operations. First, triangularize decomposes the common roots of any polynomial set into regular chains. The operations NormalForm and Inverse compute the normal form and the inverse (when it exists) of a polynomial w.r.t. a regular chain. The operation RegularGcd computes the gcd of two polynomials modulo a regular chain and MatrixInverse computes the inverse (when it exists) of a polynomial matrix w.r.t. a regular chain. As we saw above, computations modulo a regular chain may lead to a case discussion. In fact, this is achieved by means of the D5 Principle [4]. It is natural to ask whether these cases can be re-combined. The operation MatrixCombine implements an algorithm that recombine these cases, when this is possible. It is a byproduct of the notion of the equiprojectable decomposition of a polynomial system introduced in [2] and provided by the operation EquiprojectableDecomposition.Conclusions. The RegularChains library provides routines for computing (polynomial GCDs, inverses, . . . ) modulo regular chains. In particular, this includes computing over towers of field extensions (algebraic or transcendental). In general, this allows computing modulo any radical polynomial ideal, since the operation triangularize can decompose any such ideal into regular chains. New developments will be included in the next release. Indeed, the work reported in [2] has led to a modular algorithm [2] for triangular decompositions of the simple roots of a polynomial system. A preliminary implementation of it shows significant improvements in running time and allows us to solve more difficult problems. AbstractThe standard approach for computing with an algebraic number is through the data of its irreducible minimal polynomial over some base field k. However, in typical tasks such as polynomial system solving, involving many algebraic numbers of high degree, following this approach will require using probably costly factorization algorithms. Della Dora, Dicrescenzo and Duval introduced "dynamic evaluation" techniques (also termed "D5 principle") [3] as a means to compute with algebraic numbers, while avoiding factorization. Roughly speaking, this approach leads one to compute over direct products of field extensions of k, instead of only field extensions.

The structure of the heme o prosthetic group of Escherichia coli quinol oxidase (cytochrome o oxidase) has been unambiguously determined by preparation and characterization of its iron-free derivative porphyrin o dimethyl ester, or dimethyl 2,7,12,18-tetramethy1-3-[ (4E,8E)-1 -hydroxy-5,9,13-trimethyltetradeca-4,8,12-trienyl] -8-vinylporphine-13,17-dipropionate. The identity of this natural porphyrin dimethyl ester was established by 'H NMR, MS, IR, and RR spectroscopies as well as by comparisons with model compounds and the closely related porphyrin a dimethyl ester. The reliability of the structure determination was further strengthened by the isolation and characterization of the acetylated and dehydrated derivatives of porphyrin 0. ( 5 ) (a) Timkovich, R.; Cork, M. S.; Gennis, R. G.; Johnson, P. Y. J. Am.

In this paper we extend complex homotopy methods to finding witness points on the irreducible components of real varieties. In particular we construct such witness points as the isolated real solutions of a constrained optimization problem.First a random hyperplane characterized by its random normal vector is chosen. Witness points are computed by a polyhedral homotopy method. Some of them are at the intersection of this hyperplane with the components. Other witness points are the local critical points of the distance from the plane to components. A method is also given for constructing regular witness points on components, when the critical points are singular.The method is applicable to systems satisfying certain regularity conditions. Illustrative examples are given. We show that the method can be used in the consistent initialization phase of a popular method due to Pryce and Pantelides for preprocessing differential algebraic equations for numerical solution.

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