Parallel computation of real solving bivariate polynomial systems by zero-matching method

We present a new algorithm for solving the real roots of a bivariate polynomial system $\Sigma={f(x,y),g(x,y)}$ with a finite number of solutions by using a zero-matching method. The method is based on a lower bound for bivariate polynomial system when the system is non-zero. Moreover, the multiplicities of the roots of $\Sigma=0$ can be obtained by a given neighborhood. From this approach, the parallelization of the method arises naturally. By using a multidimensional matching method this principle can be generalized to the multivariate equation systems.

💡 Research Summary

The paper introduces a novel algorithm for computing all real solutions of a bivariate polynomial system Σ = {f(x,y), g(x,y)} that has a finite number of solutions. The core idea is a “zero‑matching” method that relies on a provable lower bound for the product |f·g| when the system is non‑zero, and it naturally lends itself to parallel execution.

1. Lower‑bound based pruning
The authors first partition the rectangular domain D =