Elimination Without Eliminating: Computing Complements of Real Hypersurfaces Using Pseudo-Witness Sets

Many hypersurfaces in algebraic geometry, such as discriminants, arise as the projection of another variety. The real complement of such a hypersurface partitions its ambient space into open regions. In this paper, we propose a new method for computing these regions. Existing methods for computing regions 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 can be done using so-called pseudo-witness sets without computing a defining equation for the hypersurface - we perform elimination without actually eliminating. We implement our approach in a forthcoming Julia package and demonstrate, on several examples, that the resulting algorithm accurately recovers all regions of the real complement of a hypersurface.