Detecting Simultaneous Integer Relations for Several Real Vectors

TL;DR

This paper introduces an efficient algorithm for detecting simultaneous integer relations among multiple real vectors, with proven complexity bounds and practical improvements over existing methods, including applications to algebraic number theory.

Contribution

The paper presents a novel algorithm that efficiently finds or proves the absence of small integer relations among multiple vectors, improving upon previous algorithms in both theory and practice.

Findings

Algorithm has complexity ${ m O}(n^4 + n^3 ext{log} \lambda(X))$

Experimental results show better performance than existing algorithms

Application to minimal polynomial detection is faster than Maple's built-in function

Abstract

An algorithm which either finds an nonzero integer vector ${\mathbf m}$ for given $t$ real $n$-dimensional vectors ${\mathbf x}_1,...,{\mathbf x}_t$ such that ${\mathbf x}_i^T{\mathbf m}=0$ or proves that no such integer vector with norm less than a given bound exists is presented in this paper. The cost of the algorithm is at most ${\mathcal O}(n^4 + n^3 \log \lambda(X))$ exact arithmetic operations in dimension $n$ and the least Euclidean norm $\lambda(X)$ of such integer vectors. It matches the best complexity upper bound known for this problem. Experimental data show that the algorithm is better than an already existing algorithm in the literature. In application, the algorithm is used to get a complete method for finding the minimal polynomial of an unknown complex algebraic number from its approximation, which runs even faster than the corresponding \emph{Maple} built-in function.