A complete algorithm to find exact minimal polynomial by approximations

TL;DR

This paper introduces a comprehensive algorithm that accurately derives the exact minimal polynomial of an algebraic number from its approximation, combining numerical efficiency with symbolic precision.

Contribution

It presents a novel parameterized integer relation method that improves error control and efficiency in computing exact minimal polynomials from approximations.

Findings

Superior error control in obtaining exact rational numbers from approximations

Efficient conversion from rational approximation to minimal polynomial

Effective polynomial factorization for multivariate polynomials

Abstract

We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on obtaining an exact rational number from its approximation. The algorithm is applicable for finding exact minimal polynomial of an algebraic number 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 the subsistent methods, our method combines advantage of high efficiency in numerical computation, and exact, stable results in symbolic computation. we also discuss some applications to some transcendental numbers by approximations. Moreover, the Digits of our algorithm is far less than the LLL-lattice basis reduction technique in theory. In this paper, we completely implement how to obtain exact results by numerical approximate computations.