On the Enumeration of Symmetric Tridiagonal Matrices with prescribed Determinant over Commutative Finite Chain Rings

Abstract

Determinants of structured matrices play a fundamental role in both pure and applied mathematics, with wide-ranging applications in linear algebra, combinatorics, coding theory, and numerical analysis. In this work, the enumeration of symmetric tridiagonal matrices with prescribed determinant over finite fields and over commutative finite chain rings is developed. Using the recurrent formula for determinants, a recursive form of the numbers of singular and nonsingular symmetric tridiagonal matrices is derived, after which a uniform counting framework for any fixed determinant value is obtained. Over finite fields, quadratic-character techniques are employed. In odd dimensions, the enumeration is shown to be independent of the chosen nonzero determinant. Whereas, in even dimensions, it depends only on the quadratic residue/nonresidue class in the fields. For commutative finite chain rings, explicit formulas for nonsingular case are produced by lifting along the ideal chain and analyzing reduction to the residue field, yielding closed expressions in terms of the nilpotency index and the size of the residue fields. A layered enumeration has been developed for the study of symmetric tridiagonal matrices over commutative finite chain rings by stratifying determinants into ideal and punctured layers. Entry-wise reduction to quotient rings expresses each layer through zero-determinant counts on quotients, yielding formulas for prescribed determinants, including quadratic/non-quadratic residue factors, and a decomposition for the singular ones.