Title: Numerical Elimination and Parameter Space Exploration

Speaker: Oskar Henriksson

**Parametric polynomial systems** are ubiquitous in applied mathematics, and one is often interested in understanding how the number of real or positive solutions varies across the parameter space of a model. A key tool for studying such questions is the **discriminant variety**, which partitions the parameter space into regions with constant root counts. Symbolic techniques such as cylindrical algebraic decomposition (CAD) can then be used to obtain a complete characterization of these regions.

However, in practice these symbolic approaches are often computationally very expensive. Even obtaining a defining equation for the discriminant variety via symbolic elimination can be highly challenging or infeasible for systems with only two or three variables, let alone computing a full CAD.

When symbolic methods fail, we can instead turn to numerical ones. In this talk I will present a numerical approach to analyzing discriminant varieties and their complements that uses tools from **Numerical Algebraic Geometry** to circumvent the need for symbolic elimination and CAD. The key ingredients are *witness sets* (which represent a discriminant through its intersection with a generic line); *numerical irreducible decomposition* (which identifies which witness points belong to which component of the discriminant); and *gradient roadmaps* (which represents the complement through critical points of a certain Morse function that can be evaluated using witness set data).

In my talk, I will give a brief introduction to these tools, explain how they are combined in our Julia package `ProjectedHypersurfaceRegions.jl`, and demonstrate how the package can be used to analyze the parameter spaces of some popular polynomial models in mathematical biology.

This is joint work with Paul Breiding, John Cobb, Aviva Englander, Nayda Farnsworth, Jonathan Hauenstein, David Johnson, Jordy Lopez Garcia, and Deepak Mundayur.

Paper: https://arxiv.org/pdf/2601.04383.

Code: https://github.com/oskarhenriksson/ProjectedHypersurfaceRegions.jl (under development).