A Maple Implementation for Deterministically Certifying Isolated Simple Zeros of Over-Determined Polynomial Systems with Interval Arithmetic and Its Applications

This paper presents a Maple implementation of an interval verification method for identifying isolated simple zeros in square polynomial systems. Compared to the known MATLAB (R2019b) implementation, the Maple-based approach achieves significantly higher numerical accuracy. The implementation enables polynomial evaluation at specific points to yield results with very small absolute values—sufficiently precise to reach error bounds computed through theoretical formulations for moderate-sized systems. This advancement allows the deterministic certification of isolated simple zeros in over-determined polynomial systems containing approximately 10,000 complex zeros. As a practical demonstration, the method is further applied to rigorously verify isolated multiple zeros in smaller-scale polynomial systems.