This paper presents Newton, a branch & prune algorithm to find all isolated solutions of a system of polynomial constraints. Newton can be characterized as a global search method which uses intervals for numerical correctness and for pruning the search space early. The pruning in Newton consists in enforcing at each node of the search tree a unique local consistency condition, called box-consistency, which approximates the notion of arc-consistency well-known in artificial intelligence. Box-consistency is parametrized by an interval extension of the constraint and can be instantiated to produce the Hansen-Segupta's narrowing operator (used in interval methods) as well as new operators which are more effective when the computation is far from a solution. Newton has been evaluated on a variety of benchmarks from kinematics, chemistry, combustion, economics, and mechanics. On these benchmarks, it outperforms the interval methods we are aware of and compares well with state-of-the-art continuation methods. Limitations of Newton (e.g., a sensitivity to the size of the initial intervals on some problems) are also discussed. Of particular interest is the mathematical and programming simplicity of the method.

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We propose a new and efficient algorithm for computing the sparse resultant of a system of n + 1 polynomial equations in n unknowns. This algorithm produces a matrix whose entries are coefficients of the given polynomials and is typically smaller than the matrices obtained by previous approaches. The matrix determinant is a nontrivial multiple of the sparse resultant from which the sparse resultant itself can be recovered. The algorithm is incremental in the sense that successively larger matrices are constructed until one is found with the above properties. For multigraded systems, the new algorithm produces optimal matrices, i.e., expresses the sparse resultant as a single determinant. An implementation of the algorithm is described and experimental results are presented. In addition, we propose an efficient algorithm for computing the mixed volume of n polynomials in n variables. This computation provides an upper bound on the number of common isolated roots. A publicly available implementation of the algorithm is presented and empirical results are reported which suggest that it is the fastest mixed volume code to date.

The last decade has witnessed the rebirth of resultant methods as a powerful computational tool for variable elimination and polynomial system solving. In particular, the advent of sparse elimination theory and toric varieties has provided ways to exploit the structure of polynomials encountered in a number of scientific and engineering applications. On the other hand, the Bezoutian reveals itself as an important tool in many areas connected to elimination theory and has its own merits, leading to new developments in effective algebraic geometry. This survey unifies the existing work on resultants, with emphasis on constructing matrices that generalize the classic matrices named after Sylvester, Bézout and Macaulay. The properties of the different matrix formulations are presented, including some complexity issues, with an emphasis on variable elimination theory. We compare toric resultant matrices to Macaulay's matrix and further conjecture the generalization of Macaulay's exact ratio...

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PHoM is a software package in C++ for finding all isolated solutions of polynomial systems using a polyhedral homotopy continuation method. Among three modules constituting the package, the first module StartSystem constructs a family of polyhedral-linear homotopy functions, based on the polyhedral homotopy theory, from input data for a given system of polynomial equations f (x) = 0. The second module CMPSc traces the solution curves of the homotopy equations to compute all isolated solutions of f (x) = 0. The third module Verify checks whether all isolated solutions of f (x) = 0 have been approximated correctly. We describe numerical methods used in each module and the usage of the package. Numerical results to demonstrate the performance of PHoM include some large polynomial systems that have not been solved previously.

Sparse elimination exploits the structure of a multivariate polynomial by considering its Newton polytope instead of its total degree. We concentrate on polynomial systems that generate zero-dimensional ideals. A monomial basis for the coordinate ring is defined from a mixed subdivision of the Minkowski sum of the Newton polytopes. We offer a new and simple proof relying on the construction of a sparse resultant matrix, which leads to the computation of a multiplication map and all common zeros. The size of the monomial basis equals the mixed volume and its computation is equivalent to computing the mixed volume, so the latter is a measure of intrinsic complexity. On the other hand, our algorithms have worst-case complexity proportional to the volume of the Minkowski sum. In order to derive bounds in terms of the sparsity parameters, we establish new bounds on the Minkowski sum volume as a function of mixed volume. To this end, we prove a lower bound on mixed volume in terms of euclidea...

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.

A classical circuit-design problem from Ebers and Moll [6] features a system of nine nonlinear equations in nine variables that is very challenging both for local and global methods. This system was solved globally using an interval method by Ratschek and Rokne [23] in the box [0; 10] 9 . Their algorithm had enormous costs (i.e., over 14 months using a network of 30 Sun Sparc-1 workstations) but they state that "at this time, we know no other method which has been applied to this circuit design problem and which has led to the same guaranteed result of locating exactly one solution in this huge domain, completed with a reliable error estimate." The present paper gives a novel branch-and-prune algorithm that obtains a unique safe box for the above system within reasonable computation times. The algorithm combines traditional interval techniques with an adaptation of discrete constraint-satisfaction techniques to continuous problems. Of particular interest is the simplicity o...

We study two important, and often complementary, issues in the implementation of geometric algorithms, namely exact arithmetic and degeneracy. We focus on integer arithmetic and propose a general and efficient method for its implementation based on modular arithmetic. We suggest that probabilistic modular arithmetic may be of wide interest, as it combines the advantages of modular arithmetic with randomization in order to speed up the lifting of residues to an integer. We derive general error bounds and discuss the implementation of this approach in our general-dimension convex hull program. The use of perturbations as a method to cope with input degeneracy is also illustrated. We present the implementation of a computationally efficient scheme that, moreover, greatly simplifies the task of programming. We concentrate on postprocessing, often perceived as the Achilles' heel of perturbations. Starting in the context of a specific application in robotics, we examine the complexity of p...