Counting finite index subrings of $\mathbb Z^n$

Lower bounds for the number of subrings in $\mathbb{Z}^n$

“…The authors of [9] and [1] prove that f n (p e ) is polynomial in p when e ≤ 8 by providing exact formulas for the number of irreducible subrings with each possible diagonal and then using the recurrence relation from Proposition 3.3.…”

“…Let Col(A) denote the Z-column span of the matrix A. For each α ∈ C n,e , the authors of [9] and [1] determine formulas for g α (p) by considering all closure conditions v i • v j ∈ Col(A) explicitly in order to find conditions on the variables a ij . In general, finding all the closure conditions for a given diagonal can be tedious.…”

“…From here, we row reduce and we note that finding conditions on a ij so that v i • v j is in the Z-span of the first n − 1 columns is equivalent to finding conditions on a ij so that the entries in the last column are in Z. Thus we are counting the number of solutions to expressions of the form (1) h…”

“…The answer to Question 7.1(1) is no. For example, Atanasov et al [1] show that g 5 (p 7 ) is a polynomial of degree 4, with the main term coming from the compositions (3, 2, 1, 1) and (2, 3, 1, 1). It is unclear how often pairs n and e are counterexamples to Question 7.1 (1).…”

“…For each fixed n ≤ 4, comparing the p −es coefficients of the expression in Theorem 1.1 shows that f n (p e ) is a polynomial in p. Liu [9] gives explicit formulas for f n (p e ) for e ≤ 5, which are all polynomial in p. His formula for e = 5 has a small error than Atanasov, Kaplan, Krakoff, and Menzel [1] correct. Further, the authors extend this work by giving explicit polynomial formulas for f n (p e ) when e ∈ {6, 7, 8}.…”