Simultaneous Integer Relation Detection and Its an Application

TL;DR

This paper introduces an efficient algorithm SIRD for detecting simultaneous integer relations among real vectors, with applications to finding minimal polynomials of algebraic numbers, outperforming existing methods.

Contribution

The paper presents a novel algorithm SIRD for simultaneous integer relation detection, generalizes it to complex numbers, and applies it to algebraic number minimal polynomial computation without LLL.

Findings

SIRD operates within polynomial time complexity.

Experimental results show SIRD outperforms existing algorithms.

A new method for minimal polynomial detection from approximations is proposed.

Abstract

Let $\mathbf{x_1}, ..., \mathbf{x_t} \in \mathbb{R}^{n}$. A simultaneous integer relation (SIR) for $\mathbf{x_1}, ..., \mathbf{x_t}$ is a vector $\mathbf{m} \in \mathbb{Z}^{n}\setminus\{\textbf{0}\}$ such that $\mathbf{x_i}^T\mathbf{m} = 0$ for $i = 1, ..., t$. In this paper, we propose an algorithm SIRD to detect an SIR for real vectors, which constructs an SIR within $\mathcal {O}(n^4 + n^3 \log \lambda(X))$ arithmetic operations, where $\lambda(X)$ is the least Euclidean norm of SIRs for $\mathbf{x_1}, >..., \mathbf{x_t}$. One can easily generalize SIRD to complex number field. Experimental results show that SIRD is practical and better than another detecting algorithm in the literature. In its application, we present a new algorithm for finding the minimal polynomial of an arbitrary complex algebraic number from its an approximation, which is not based on LLL. We also provide a sufficient condition on the precision of the approximate value, which depends only on the height and the degree of the algebraic number.