arXiv subjects
Publications and source records attributed to David K. Johnson.
We introduce a low-memory framework for certifying numerical solutions to polynomial systems which uses solution iterators and spatial partitioning trees to reduce memory requirements. We provide a prototypical algorithm, analyze its complexity, and demonstrate the memory reduction on a large example.
Many hypersurfaces in algebraic geometry, such as discriminants, arise as the projection of another variety. The real complement of such a hypersurface decomposes into connected components. In this paper, we propose a new method for computing these components. Existing methods require the explicit equation of the hypersurface as input. However, computing this equation by elimination can be computationally demanding or even infeasible. Our approach instead derives from univariate interpolation by computing the intersection of the hypersurface with a line. Such an intersection may be computed using so-called pseudo-witness sets without computing a defining equation for the hypersurface. We implement our approach in a forthcoming Julia package and demonstrate, on several examples, that the resulting algorithm accurately recovers all components of the real complement of the hypersurface.