2020/05/01 by Sarthak Chimni, Chimni, Sarthak, Ramin Takloo-Bighash +1
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2005.00150
openalex publication_date 2020/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this note we study the distribution of the subrings of \mathbb Z[t]/(t4) and prove two results. The first result gives an asymptotic formula for the number of subrings of \mathbb Z[t]/(t4) of bounded index. The method of proof of this theorem is p-adic integration a la Grunewald, Segal, and Smith. Our second result is about the distribution of cocyclic subrings in \mathbb Z[t]/(t4). Our proof of this result is combinatorial and is based on counting certain classes of matrices with Smith normal forms of a special form.