In engineering and applied mathematics, polynomial systems arise whose solution sets contain components of different dimensions and multiplicities. In this article we present algorithms, based on homotopy continuation, that compute much of the geometric information contained in the primary decomposition of the solution set. In particular, ignoring multiplicities, our algorithms lay out the decomposition of the set of solutions into irreducible components, by finding, at each dimension, generic points on each component. As by-products, the computation also determines the degree of each component and an upper bound on itsmultiplicity. The bound issharp (i.e., equal to one) for reduced components. The algorithms make essential use of generic projection and interpolation, and can, if desired, describe each irreducible component precisely as the common zeroesof a finite number of polynomials.

Many applications modeled by polynomial systems have positive dimensional solution components (e.g., the path synthesis problems for four-bar mechanisms) that are challenging to compute numerically by homotopy continuation methods. A procedure of A. Sommese and C. Wampler consists in slicing the components with linear subspaces in general position to obtain generic points of the components as the isolated solutions of an auxiliary system. Since this requires the solution of a number of larger overdetermined systems, the procedure is computationally expensive and also wasteful because many solution paths diverge. In this article an embedding of the original polynomial system is presented, which leads to a sequence of homotopies, with solution paths leading to generic points of all components as the isolated solutions of an auxiliary system. The new procedure significantly reduces the number of paths to solutions that need to be followed. This approach has been implemented and applied to...

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Homotopy continuation methods have proven to be reliable and efficient to approximate all isolated solutions of polynomial systems. In this paper we show how we can use this capability as a blackbox device to solve systems which have positive dimensional components of solutions. We indicate how the software package PHCpack can be used in conjunction with Maple and programs written in C. We describe a numerically stable algorithm for decomposing positive dimensional solution sets of polynomial systems into irreducible components.

We report on some experience with a new version of the well known Gröbner algorithm with factorization and constraint inequalities, implemented in our REDUCE package CALI, [12]. We discuss some of its details and present run time comparisons with other existing implementations on well splitting examples.

There are discussed implementational aspects of the special-purpose computer algebra system FELIX designed for computations in constructive algebra. In particular, data types developed for the representation of and computation with commutative and non-commutative polynomials are described. Furthermore, comparisons of time and memory requirements of di erent polynomial representations are reported. 1

In memoriam to Renate. We report on some experiences with the general purpose Computer Algebra Systems (CAS) Axiom, Macsyma, Maple, Mathematica, MuPAD, and Reduce solving systems of polynomial equations and the way they present their solutions. This snapshot (taken in spring 1996) of the current power of the different systems in a special area concentrates both on CPU-times and the quality of the output. 1

We describe an algorithm which computes all subfields of an effectively given finite algebraic extension. Although the base field can be arbitrary, we focus our attention on the rationals. This appears to be a fundamental tool for the simplification of algebraic numbers. Introduction Many algorithms in computer algebra contain subroutines which require to use algebraic numbers. Computing with them is especially important when polynomial systems of equation have to be solved. As an example let us consider the now called cyclic 7th--roots of unit, which are the solutions of the following system [4, 1] : a + b + c + d + e + f + g = 0 ab + bc + cd + de + ef + fg + ga = 0 abc + bcd + cde + def + efg + fga + gab = 0 abcd + bcde + cdef + defg + efga + fgab + gabc = 0 abcde + bcdef + cdefg + defga + efgab + fgabc + gabcd = 0 abcdef + bcdefg + cdefga + defgab + efgabc + fgabcd + gabcde = 0 abcdefg = 1: Some of the solutions of this system are of the form (a; b; c; 1=c; 1=b; 1=a; 1...