Detecting Simultaneous Integer Relations for Several Real Vectors

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.

💡 Research Summary

The paper addresses the problem of detecting simultaneous integer relations (SIRs) among several real vectors. Given t real n‑dimensional vectors (x_1,\dots,x_t) with (t<n) and linearly independent, an SIR is a non‑zero integer vector (m\in\mathbb Z^n) such that (x_i^{!T}m=0) for all i. The authors propose an algorithm that either finds such an (m) or proves that no integer relation with Euclidean norm below a prescribed bound exists.

The core of the method is the construction of a hyperplane matrix (H\in\mathbb R^{n\times (n-t)}) whose columns span the orthogonal complement of the space generated by the input vectors. This matrix is built by applying a Gram–Schmidt process to the concatenation ((x_1,\dots,x_t,b_1,\dots,b_n)) where (b_i) are the standard basis vectors. Lemma 2.2 guarantees that (H) is lower‑trapezoidal, has non‑zero diagonal entries, satisfies (X^{!T}H=0) and (H^{!T}H=I_{n-t}), and its Frobenius norm equals (\sqrt{n-t}).

A novel “generalized Hermite reduction” is introduced to reduce (H). While the classic modified Hermite reduction (used in PSLQ) only touches the first (n-t) rows, the generalized version also processes the last (t-1) rows, ensuring that the reduction matrix (D) remains unimodular ((D\in GL(n,\mathbb Z))) and that each off‑diagonal entry is at most half the magnitude of the corresponding diagonal entry. Row exchanges are performed when certain trailing entries become zero, preserving the lower‑trapezoidal shape.

Algorithm 1 proceeds as follows:

1. Initialization – compute (H) and set (B=I_n); apply the generalized Hermite reduction to obtain matrices (D), update (X^{!T}\leftarrow X^{!T}D^{-1}), (H\leftarrow DH), (B\leftarrow BD^{-1}).
2. Iteration – repeat four sub‑steps:
   • Exchange – select an index (r) maximizing (\gamma|h_{r,r}|) (with (\gamma>2/\sqrt3)) and swap rows (r) and (r+1) via a permutation matrix (R).
   • Corner – if (r<n-t), apply a 2×2 orthogonal transformation (Q) to the sub‑matrix formed by rows/columns (r,r+1), thereby reducing the size of the off‑diagonal entries.
   • Reduction – run the generalized Hermite reduction again, updating (X^{!T}, H, B).
   • Termination test – compute (G=1/|H|F). If any column of (B) satisfies (X^{!T}B_j=0) or if the last diagonal entry (h{n-t,n-t}=0), output the corresponding column as an SIR. Otherwise, if no SIR has norm < (G), declare that no such relation exists.

Theoretical analysis (Theorem 2.8) shows that the algorithm terminates after at most \