Exact Bivariate Polynomial Factorization in Q by Approximation of Roots

Factorization of polynomials is one of the foundations of symbolic computation. Its applications arise in numerous branches of mathematics and other sciences. However, the present advanced programming languages such as C++ and J++, do not support symbolic computation directly. Hence, it leads to difficulties in applying factorization in engineering fields. In this paper, we present an algorithm which use numerical method to obtain exact factors of a bivariate polynomial with rational coefficients. Our method can be directly implemented in efficient programming language such C++ together with the GNU Multiple-Precision Library. In addition, the numerical computation part often only requires double precision and is easily parallelizable.

💡 Research Summary

The paper addresses a long‑standing challenge in computer algebra: obtaining exact factorizations of multivariate polynomials with rational coefficients using only numerical tools that can be efficiently implemented in high‑performance languages such as C++. Traditional symbolic systems (Maple, Mathematica, etc.) provide exact factorization but suffer from high memory consumption and limited scalability, especially for high‑degree bivariate polynomials. Moreover, modern engineering software stacks rarely expose symbolic capabilities, creating a gap between theoretical algebraic methods and practical engineering applications.

The authors propose a hybrid algorithm that combines a purely numerical root‑approximation phase with a rigorous rational reconstruction phase. The method works for any bivariate polynomial f(x, y) ∈ Q