2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/139869The principal minors of a symmetric $n{\times}n$-matrix form a vector of length $2^n$. We characterize these vectors in terms of algebraic equations derived from the $ 2{\times}2{\times}2$-hyperdeterminant.12 pages; referee's suggestions incorporated, new section "Invariance and the Quartic Generation Conjecture" addedRings and AlgebrasCommutative Algebra14M12, 15A15, 15A72, 15A29, 13P10text