Numerical approach for solving problems arising from polynomial analysis
Yousra Gati, Vladimir Petrov Kostov, Mohamed Chaouki Tarchi
TL;DR
This work addresses realizability questions for polynomials under Descartes' rule of signs and for hyperbolic polynomials with prescribed moduli orders, by deploying a numerical framework that samples random roots to construct candidate polynomials. The approach yields concrete realizations in several cases, extends analytic methods like the concatenation lemma, and provides numerical evidence toward conjectures on distances between zeros and critical points within the Laguerre-Pólya class $\mathcal{LP}1$. It demonstrates that numerical root-sampling is a practical, fast tool to guide theoretical proofs, test boundary cases, and identify likely non-realizable configurations, thereby supporting and accelerating research in polynomial analysis. The results include explicit polynomials realizing various sign-pattern and modulus-order combinations up to degrees $7$ and $6$ and a degree-$6$ example supporting a subtle case of the conjectured inequalities. Overall, the method complements analytic techniques, offering a robust computational aid for exploring realizability and related polynomial properties.
Abstract
This paper deals with the use of numerical methods based on random root sampling techniques to solve some theoretical problems arising in the analysis of polynomials. These methods are proved to be practical and give solutions where traditional methods might fall short.