Edge reduction for unit forms
We give an algorithm which decides whether or not an integral quadratic form \(q: \mathbb{Z}^n\to \mathbb{Z}\) of the form \(q(x)= \sum_i x_i^2+ \sum_{i<j} q_{ij} x_i x_j\) (unit form) is weakly positive, that is, satisfies \(q(x)> 0\) for all \(0\neq x\in \mathbb{N}^n\); if this is the case we obtain an explicit description of the set \(R_q= \{x\in \mathbb{N}^n\mid q(x)= 1\}\) of positive roots of \(q\). The presented algorithm is an adaptation of the `edge reduction' algorithm formulated originally by \textit{A. V. Roiter} and \textit{M. M. Kleiner} for DGC's and bocses [Lect. Notes Math. 488, 316-339 (1975; Zbl 0356.16011)].
scientific article; zbMATH DE number 815693