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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.21101 |
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Table of Contents:
- Saito's criterion is a foundational result that algebraically characterizes free hyperplane arrangements via the determinant of a square matrix of logarithmic derivations. It is natural to ask whether this criterion can be generalized to the non-free setting. To address this, we formulate a general problem concerning the maximal minors of a $p \times \ell$ ($p \geq \ell$) derivation matrix and the algebraic relations among their associated coefficients. Focusing on strictly plus-one generated (SPOG) arrangements, we completely solve this minor-based recognition problem under the assumption that $\operatorname{pd} D(\mathcal{A}) \leq 1$. As a direct consequence, we obtain a purely algebraic, necessary and sufficient characterization of SPOG arrangements in dimension three. Ultimately, this framework provides a computable bridge to post-free arrangement theory.