[Paper Review] Counting ideals in polynomial rings

Lenny Fukshansky, Stefan Kühnlein+1 more|arXiv (Cornell University)|Jan 17, 2017

Coding theory and cryptography5 references3 citations

TL;DR

This paper investigates the asymptotic growth of ideal counting functions in polynomial rings over rings of integers in number fields, using zeta functions and Delange's Tauberian theorem. It establishes that the number of ideals of bounded index in $\mathcal{O}_K[X]$ grows asymptotically as $O(N^2)$, generalizing classical results on Dedekind zeta functions and providing density estimates for ideal lattices in $\mathbb{Z}^d$. The key contribution is a closed-form product formula for the zeta function of $\mathcal{O}_K[X]$, revealing its meromorphic structure and pole at $s=2$.

ABSTRACT

We investigate properties of zeta functions of polynomial rings and their quotients, generalizing and extending some classical results about Dedekind zeta functions of number fields. By an application of Delange's version of the Ikehara Tauberian Theorem, we are then able to determine the asymptotic order of the ideal counting function in such rings. As a result, we produce counting estimates on ideal lattices of bounded determinant coming from fixed number fields, as well as density estimates for any ideal lattices among all sublattices of $\mathbb Z^d$. We conclude with some more general speculations and open questions.

Motivation & Objective

• Understand the asymptotic behavior of the number of ideals of bounded index in polynomial rings over rings of integers in number fields.
• Generalize classical results on Dedekind zeta functions to polynomial rings $\mathcal{O}_K[X]$.
• Establish counting estimates for ideal lattices of bounded determinant arising from fixed number fields.
• Analyze the analytic properties of zeta functions associated with quotient rings $\mathbb{Z}[X]/(f)$ for monic separable polynomials $f$.
• Provide density estimates for ideal lattices among all sublattices of $\mathbb{Z}^d$.
• investigate the structure of zeta functions for $\mathcal{O}_K[X]$ and their meromorphic extensions.
• Explore connections to abelian group classification and Cohen-Lenstra heuristics via zeta functions of $\mathbb{Z}[X]$.

Proposed method

• The paper employs zeta functions $\zeta(R,s) = \sum_{n=1}^\infty \frac{a_n(R)}{n^s}$, where $a_n(R)$ counts ideals of index $n$ in a ring $R$, to study ideal counting.
• Delange's version of the Ikehara Tauberian Theorem is applied to deduce asymptotic growth of the summatory function $A_N(R) = \sum_{n \leq N} a_n(R)$ from the analytic behavior of $\zeta(R,s)$.
• The zeta function of $\mathcal{O}_K[X]$ is shown to factor as $\prod_{d=1}^\infty \zeta_K(d(s-1))$, where $\zeta_K$ is the Dedekind zeta function of the number field $K$.
• By analyzing the poles of this product, the abscissa of convergence and the location of singularities are determined, with the dominant pole at $s=2$.
• Multiplicativity of the ideal counting sequence $a_n(R)$ is used to derive an Euler product decomposition $\zeta(R,s) = \prod_p E_p(R,s)$, enabling local analysis at each prime.
• Local zeta functions are studied via base change to $\mathbb{Z}_{(p)} \otimes_{\mathbb{Z}} \mathcal{O}_K$, which is a semilocal PID, allowing the use of known results on PIDs to derive the global product formula.
• Connections to abelian group classification are drawn by identifying $\sum a_n(\mathfrak{G})/n^s = \zeta(\mathbb{Z}[X], s+1)$, linking ideal counting to group-theoretic orbit counts.

Experimental results

Research questions

• RQ1 What is the asymptotic growth rate of the number of ideals of bounded index in the polynomial ring $\mathcal{O}_K[X]$ over the ring of integers of a number field $K$?
• RQ2 How does the zeta function $\zeta(\mathcal{O}_K[X], s)$ behave analytically, and what is its meromorphic structure?
• RQ3 What is the precise asymptotic order of the number of ideal lattices of bounded determinant in $\mathbb{Z}^d$ arising from quotient rings $\mathbb{Z}[X]/(f)$ for monic polynomials $f$?
• RQ4 How do ideal lattices in $\mathbb{Z}^d$ compare in density to all sublattices of $\mathbb{Z}^d$?
• RQ5 What is the relationship between the zeta function of $\mathbb{Z}[X]$ and the enumeration of finite abelian groups or their subgroups under automorphism actions?
• RQ6 Can the zeta function of $\mathcal{O}_K[X]$ be interpreted geometrically as a zeta function of a deformation of the affine line over $\mathcal{O}_K$?
• RQ7 Is there a natural interpretation of $\zeta(\mathbb{Z}[X], s)$ as counting finite abelian groups with an endomorphism satisfying certain properties?

Key findings

• The zeta function of the polynomial ring $\mathcal{O}_K[X]$ is given by $\zeta(\mathcal{O}_K[X], s) = \prod_{d=1}^\infty \zeta_K(d(s-1))$, where $\zeta_K$ is the Dedekind zeta function of the number field $K$.
• The zeta function $\zeta(\mathcal{O}_K[X], s)$ has an abscissa of convergence at $\sigma = 2$ and a simple pole at $s = 2$, with additional simple poles accumulating at $s = 1$.
• Applying Delange's Tauberian Theorem, the number of ideals in $\mathcal{O}_K[X]$ of index at most $N$ grows asymptotically as $c \cdot N^2$ for some constant $c > 0$, with $c$ depending on the residue of $\zeta(\mathcal{O}_K[X], s)$ at $s = 2$.
• The number of ideal sublattices of $\mathbb{Z}^d$ arising from quotient rings $\mathbb{Z}[X]/(f)$ for monic polynomials $f$ of degree $d$ also grows asymptotically as $O(N^2)$, as shown in Corollary 3.3.
• The zeta function $\zeta(\mathbb{Z}[X], s)$ converges for $\Re(s) > 2$ and has a simple pole at $s = 2$, confirming the quadratic growth of ideal counting in $\mathbb{Z}[X]$.
• The zeta function $\zeta(\mathbb{Z}[X], s)$ is related to the enumeration of finite abelian groups: $\sum_{n=1}^\infty \frac{a_n(\mathfrak{G})}{n^s} = \zeta(\mathbb{Z}[X], s+1)$, where $a_n(\mathfrak{G})$ counts isomorphism classes of abelian groups of order $n$.
• The residue of $\zeta(\mathcal{O}_K[X], s)$ at $s = 2$ is identified with the constant $C_\infty$ appearing in Cohen-Lenstra heuristics, suggesting a deep connection between ideal counting and random finite abelian groups.