Fast and accurate solver for high-degree univariate resultant determinant

Develop a computationally efficient and numerically stable algorithm to solve the univariate equation r(v1) = det R(v1) = 0, where R(v1) is the Be
7zout resultant matrix obtained by eliminating one variable from the bivariate specular polynomial system. The solver should handle arbitrarily high polynomial degrees and reliably find all zeros needed to recover admissible specular paths in the authors' specular polynomial framework.

Background

In the proposed pipeline, multivariate specular constraints are reduced to bivariate polynomials, and then to a univariate problem via the hidden variable resultant method: r(v1) = det R(v1) = 0. For one-bounce cases, explicit coefficient computation followed by Laplacian expansion and bisection can be practical; for longer chains, however, constructing and solving large determinant polynomials becomes challenging.

The authors note that while linearization via a generalized eigenvalue problem and QZ decomposition is theoretically comprehensive, its computational burden (with time complexity scaling approximately as O(k n^6)) is prohibitive in practice for high-degree systems. Their piecewise bisection approach works well empirically but can miss clustered roots and lacks the theoretical guarantees of eigenvalue methods.

Consequently, an efficient, accurate, and scalable solver for the univariate determinant equation arising from the resultant r(v1) remains a bottleneck. A solution to this problem would directly strengthen the specular polynomial pipeline by enabling robust handling of longer specular chains and higher-degree polynomial systems.