Closed form for zero-determinant enumeration on quotient levels of CFCRs

Establish closed-form expressions for |ST_n(R/γ^t R, 0)|, the number of n×n symmetric tridiagonal matrices over the quotient ring R/γ^t R whose determinant is zero, for arbitrary positive integers n and t, where R is a commutative finite chain ring with maximal ideal generated by γ.

Background

The paper develops a layered enumeration technique over commutative finite chain rings (CFCRs), reducing counts over R to counts over its quotients R/γ^t R. For nonsingular cases, explicit formulas are obtained by lifting from the residue field. For singular cases, the authors express counts in terms of zero-determinant counts on quotient rings.

A central remaining step for fully explicit formulas is obtaining closed forms for the number of symmetric tridiagonal matrices with determinant zero over each quotient level R/γ^t R. This would complete the layered enumeration by enabling direct computation of |ST_n(R,0)| via the established reduction identities.