Singular Cholesky Fibers over Finite Fields

Abstract: For a finite field $\F_q$, consider the triangular Cholesky map $Γ<em>{n,q}(U)=U<sup>TU$ from upper triangular matrices to symmetric matrices. Generalized Cholesky theory describes the regular locus on which all leading principal minors are nonzero, but it does not determine the multiplicities or root ranks in a fiber over a singular target. We develop a fixed-target fiber theory for this singular boundary and answer three questions posed by Cooper and Whitlatch. Our principal results concern the zero fiber. Over $\F_2$ we construct an explicit, invertible, rank-preserving recursive bijection between square-zero upper triangular matrices and upper triangular matrices satisfying $U<sup>TU=0$. Over every finite field we determine the entire rank distribution of this zero fiber: in even characteristic the square-zero recurrence, and hence the rank-refined equinumerosity, persists over every $\F</em>{2<sup>e}$, whereas in odd characteristic its failure is measured by an explicit quadratic-character correction. We place these results in a uniform framework by proving an exact first-pivot recursion for the fiber cardinality over an arbitrary symmetric target. Its regular specialization gives the constant fiber sizes on leading-principal-minor cones, while its zero-pivot branch explains the new singular behavior. For binary diagonal targets we further compress the rank-refined count to $O(n<sup>2)$ integer-arithmetic transitions and show that it depends on the order, not merely the number, of the diagonal entries. Executable implementations and exhaustive low-order checks are collected in an appendix.