Counting Ideals in $\mathbb{Z}[t]/(f)$
In this paper we study the growth of ideals in $\mathbb{Z}[t]/(f)$ for a monic cubic polynomial $f$. We also compute the ideal zeta function of $\mathbb{Z}[t]/(t^n)$ for any $n \in \mathbb{N}$.
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arXiv subjects
Sarthak Chimni
Publications and source records attributed to Sarthak Chimni.
Cotype zeta function for subrings of ${\mathbb Z}[t]/(t^3)$
In this note we study the cotypes of subrings of ${\mathbb Z}[t]/(t^3)$ using $p$-adic integration.
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Subrings of $\mathbb Z[t]/(t^4)$
In this note we study the distribution of the subrings of $\mathbb Z[t]/(t^4)$ and prove two results. The first result gives an asymptotic formula for the number of subrings of $\mathbb Z[t]/(t^4)$ 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 $\mathbb Z[t]/(t^4)$. Our proof of this result is combinatorial and is based on counting certain classes of matrices with Smith normal forms of a special form.
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Counting subrings of $\mathbb Z^n$ of non-zero co-rank
In this paper we study subrings of $\mathbb Z^{n+k}$ of co-rank $k$.
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