Year: 2009
Type: preprint
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 $\mathcal {O}(n^4 + n^3 \log λ(X))$ arithmetic operations, where $λ(X)$ 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.
Source: arXiv (Cornell University)
Authors: Jing-wei, Chen, Yong, Feng, Xiao-lin, Qin, Jing-zhong, Zhang
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1629157255