Efficient computation of residual arrangements

Develop more efficient software or an algorithm for computing residual arrangements of positive polytopes in \(\mathrm{Gr}(2,4)\), sufficient to handle all 3360 combinatorial types of positive hexahedra, or determine whether such an improvement is possible.

Background

The paper computes residual arrangements using cylindrical algebraic decomposition in Mathematica. The procedure is sufficiently complicated and computationally expensive that the authors do not carry it out for all 3360 combinatorial types of positive hexahedra. They therefore leave unresolved whether more efficient software or algorithms can make a complete classification computationally feasible.

References

We compute residual arrangements of positive polytopes by using cylindrical algebraic decomposition in Mathematica. This is a complicated procedure which prevents us from doing the computation for all $3360$ combinatorial types of positive hexahedra. Is there more efficient software (or a more efficient algorithm) to do this? If not, can one develop such software?

— Positive Polytopes with Few Facets in the Grassmannian  (2503.01652 - Pavlov et al., 3 Mar 2025) in Question environment, Section 6, “Open questions and future directions”