Abstract. This article treats numerical methods for tracking an implicitly defined path. The numerical precision required to successfully track such a path is difficult to predict a priori, and indeed, it may change dramatically through the course of the path. In current practice, one must either choose a conservatively large numerical precision at the outset or re-run paths multiple times in successively higher precision until success is achieved. To avoid unnecessary computational cost, it would be preferable to adaptively adjust the precision as the tracking proceeds in response to the local conditioning of the path. We present an algorithm that can be set to either reactively adjust precision in response to step failure or proactively set the precision using error estimates. We then test the relative merits of reactive and proactive adaptation on several examples arising as homotopies for solving systems of polynomial equations.

This article introduces the concept of isosingular sets, which are irreducible algebraic subsets of the set of solutions to a system of polynomial equations that share a common singularity structure. The definition of these sets depends on deflation, a procedure that uses differentiation to regularize solutions. A weak form of deflation has proven useful in regularizing algebraic sets, making them amenable to treatment by the algorithms of numerical algebraic geometry. We introduce a strong form of deflation and define deflation sequences, which are similar to the sequences arising in Thom-Boardman singularity theory. We then define isosingular sets in terms of deflation sequences. We also define the isosingular local dimension and examine the properties of isosingular sets. While isosingular sets are of theoretical interest as constructs for describing singularity structures of algebraic sets, they also expand the kinds of algebraic sets that can be investigated with methods from numerical algebraic geometry.

Abstract. We present a new technique, based on polynomial continuation, for solving systems of n polynomials in N complex variables. The method allows equations to be introduced one-by-one or in groups, obtaining at each stage a representation of the solution set that can be extended to the next stage until finally obtaining the solution set for the entire system. At any stage where positive dimensional solution components must be found, they are sliced down to isolated points by the introduction of hyperplanes. By moving these hyperplanes, one may build up the solution set to an intermediate system in which a union of hyperplanes "regenerates" the intersection of the component with the variety of the polynomial (or system of polynomials) brought in at the next stage. The theory underlying the approach guarantees that homotopy paths lead to all isolated solutions, and this capability can be used to generate witness supersets for solution components at any dimension, the first step in computing an irreducible decomposition of the solution set of a system of polynomial equations. The method is illustrated on several challenging problems, where it proves advantageous over both the polyhedral homotopy method and the diagonal equation-by-equation method, formerly the two leading approaches to solving sparse polynomial systems by numerical continuation.

Abstract. Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.

Abstract. We study the geometry underlying the difference between nonnegative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

Given a polynomial system f , a fundamental question is to determine if f has real roots. Many algorithms involving the use of infinitesimal deformations have been proposed to answer this question. In this article, we transform an approach of Rouillier, Roy, and Safey El Din, which is based on a classical optimization approach of Seidenberg, to develop a homotopy based approach for computing at least one point on each connected component of a real algebraic set. Examples are presented demonstrating the effectiveness of this parallelizable homotopy based approach.

Abstract. Though numerical methods to find all the isolated solutions of nonllnestr systems of multivariate polynomials go back 30 years, it is only over the last decade that numerical methods have been devised for the computation and manipulation of algebraic sets coming from polynomJEil systems over the complex numbers. Collectively, these algorithms and the underlying theory have come to be known as numerical algebraic geometry. Several software packages are capable of carrying out some of the operations of numerical algebraic geometry, although no one package provides all such capabilities. This paper contains an enumeration of the operations that an ideal software package in this field would allow. The current and upcoming capabilities of Bertini, the most recently released package in this field, are also described.

Finding equilibria of the finite size Kuramoto model amounts to solving a nonlinear system of equations, which is an important yet challenging problem. We translate this into an algebraic geometry problem and use numerical methods to find all of the equilibria for various choices of coupling constants K, natural frequencies, and on different graphs. We note that for even modest sizes (N ∼ 10–20), the number of equilibria is already more than 100 000. We analyze the stability of each computed equilibrium as well as the configuration of angles. Our exploration of the equilibrium landscape leads to unexpected and possibly surprising results including non-monotonicity in the number of equilibria, a predictable pattern in the indices of equilibria, counter-examples to conjectures, multi-stable equilibrium landscapes, scenarios with only unstable equilibria, and multiple distinct extrema in the stable equilibrium distribution as a function of the number of cycles in the graph.