Real-rooted integer polynomial enumeration algorithms and interlacing polynomials via linear programming
Gary R. W. Greaves, Jeven Syatriadi
TL;DR
This work advances the computational enumeration of real-rooted integer polynomials by (i) extending McKee–Smyth–Smyth-style algorithms to cover reducible and multi-root cases with fixed leading coefficients, and (ii) introducing linear programming frameworks to efficiently enumerate Seidel interlacing polynomials derived from a given family of such polynomials. It provides rigorous termination guarantees and numerical-robust endpoints using EndPoints and Sturm-type checks, while delivering substantial speedups for interlacing tasks relevant to Seidel matrices and real equiangular lines. The combination of these methods – including even/odd-degree Seidel-feasible algorithms, delta-partial Seidel feasibility, and LP-based interlacing – yields a comprehensive toolkit for addressing trace-type problems and equiangular-line bounds with a focus on real-rooted, integer polynomials. Overall, the results significantly improve practical feasibility for high-degree polynomial enumeration and interlacing analyses in combinatorial and geometric contexts.
Abstract
We extend the algorithms of Robinson, Smyth, and McKee--Smyth to enumerate all real-rooted integer polynomials of a fixed degree, where the first few (at least three) leading coefficients are specified. Additionally, we introduce new linear programming algorithms to enumerate all feasible interlacing polynomials of a given polynomial that comes from a certain family of real-rooted integer polynomials. These algorithms are further specialised for the study of real equiangular lines, incorporating additional number-theoretic constraints to restrict the enumeration. Our improvements significantly enhance the efficiency of the methods presented in previous work by the authors.