We present the Julia package HomotopyContinuation.jl, which provides an algorithmic framework for solving polynomial systems by numerical homotopy continuation. We introduce the basic capabilities of the package and demonstrate the software on an illustrative example. We motivate our choice of Julia and how its features allow us to improve upon existing software packages with respect to usability, modularity and performance. Furthermore, we compare the performance of HomotopyContinuation.jl to the existing packages Bertini and PHCpack.

We establish interval arithmetic as a practical tool for certification in numerical algebraic geometry. Our software HomotopyContinuation.jl now has a built-in function certify, which proves the correctness of an isolated solution to a square system of polynomial equations. The implementation rests on Krawczyk's method. We demonstrate that it dramatically outperforms earlier approaches to certification. We see this contribution as the basis for a paradigm shift in numerical algebraic geometry where certification is the default and not just an option.

The join set of a finite collection of smooth embedded submanifolds of a mutual vector space is defined as their Minkowski sum. Join decompositions generalize some ubiquitous decompositions in multilinear algebra, namely tensor rank, Waring, partially symmetric rank and block term decompositions. This paper examines the numerical sensitivity of join decompositions to perturbations; specifically, we consider the condition number for general join decompositions. It is characterized as a distance to a set of ill-posed points in a supplementary product of Grassmannians. We prove that this condition number can be computed efficiently as the smallest singular value of an auxiliary matrix. For some special join sets, we characterized the behavior of sequences in the join set converging to the latter's boundary points. Finally, we specialize our discussion to the tensor rank and Waring decompositions and provide several numerical experiments confirming the key results.

We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.