Finding Exact Minimal Polynomial by Approximations

TL;DR

This paper introduces an efficient algorithm that reconstructs exact algebraic numbers and their minimal polynomials from approximate values, improving computational speed and accuracy over existing methods.

Contribution

The paper presents a novel parameterized integer relation method for exact polynomial reconstruction from approximations, with applications to polynomial factorization and rational number recovery.

Findings

More efficient than Maple 11's identify function

Successfully reconstructs exact algebraic numbers from approximations

Enables conversion from rational approximations to minimal polynomial representations

Abstract

We present a new algorithm for reconstructing an exact algebraic number from its approximate value using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on obtaining an exact rational number from its approximation. The algorithm is applicable for finding exact minimal polynomial by its approximate root. This also enables us to provide an efficient method of converting the rational approximation representation to the minimal polynomial representation, and devise a simple algorithm to factor multivariate polynomials with rational coefficients. Compared with other methods, this method has the numerical computation advantage of high efficiency. The experimental results show that the method is more efficient than \emph{identify} in \emph{Maple} 11 for obtaining an exact algebraic number from its approximation. In this paper, we completely implement how to obtain exact results by numerical approximate computations.