arXiv subjects

James Punch

Publications and source records attributed to James Punch.

4 recordsLinked to original sources

arXiv2025

Bounds on the Minkowski constants and a function involving $\varphi$

In 1887, Minkowski determined the least common multiple of the orders of all finite subgroups of $GL_n(\mathbb{Q})$; we refer to this number as $M(n)$. In (Katznelson, 1994), Katznelson provides the asymptotic behaviour of $M(n)$, with a small error term. In this paper, we use elementary techniques to find explicit upper and lower bounds on $M(n)$ that improve on Katznelson's results; we also recover his asymptotic result. Our results immediately imply explicit bounds on functions closely related to $M(n)$, which appear in the study of abelian varieties (see, for example, (Silverberg, 1992), (Guralnick and Kedlaya, 2017) and (Ozeki, 2024)). Finally, we examine the function $\Phi(n)$, which also appears in (Ozeki, 2024), defined as the greatest positive integer $m$ for which $\varphi(m)$ divides $2n$. We provide explicit upper bounds on $\Phi(n)$.

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arXiv2025

The modified prime sieve for primitive elements in finite fields

Let $r \geq 2$ be an integer, $q$ a prime power and $\mathbb{F}_{q}$ the finite field with $q$ elements. Consider the problem of showing existence of primitive elements in a subset $\mathcal{A} \subseteq \mathbb{F}_{q^r}$. We prove a sieve criterion for existence of such elements, dependent only on an estimate for the character sum $\sum_{\gamma \in \mathcal{A}}\chi(\gamma)$. The flexibility and direct applicability of our criterion should be of considerable interest for problems in this field. We demonstrate the utility of our result by tackling a problem of Fernandes and Reis (2021) with $\mathcal{A}$ avoiding affine hyperplanes, obtaining significant improvements over previous knowledge.

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arXiv2025

An improved upper bound on the covering radius of the logarithmic lattice of $\mathbb{Q}(\zeta_n)$

Let $\mathbb{R}^m$ be endowed with the Euclidean metric. The covering radius of a lattice $\Lambda \subset \mathbb{R}^m$ is the least distance $r$ such that, given any point of $\mathbb{R}^m$, the distance from that point to $\Lambda$ is not more than $r$. Lattices can occur via the unit group of the ring of integers in an algebraic number field $\mathbb{K}$, by applying a logarithmic embedding $\mathbb{K}^*\rightarrow \mathbb{R}^m$. In this paper, we examine those lattices which arise from the cyclotomic number field $\mathbb{Q}(\zeta_n)$, for a given positive integer $n\geq5$ such that $n\not \equiv 2\pmod{4}$. We then provide improvements to an upper bound in (de Araujo, 2024), and conclude with an upper bound on the covering radius for this lattice in terms of $n$ and the number of its distinct prime factors. In particular, we improve Lemma 2 of (de Araujo, 2024) and show that, asymptotically, it can be improved no further.

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arXiv2025

Quadratic units and cubic fields

We investigate Eisenstein discriminants, which are squarefree integers $d \equiv 5 \pmod{8}$ such that the fundamental unit $\varepsilon_d$ of the real quadratic field $K=\mathbb{Q}(\sqrt{d})$ satisfies $\varepsilon_d \equiv 1 \pmod{2\mathcal{O}_K}$. These discriminants are related to a classical question of Eisenstein and have connections to the class groups of orders in quadratic fields as well as to real cubic fields. We present numerical computations of Eisenstein discriminants up to $10^{11}$, suggesting that their counting function up to $x$ is approximated by $\pi_{\mathcal{E}}(x) \approx \frac{1}{3\pi^2}x - 0.024x^{5/6}$. This supports a conjecture of Stevenhagen while revealing a surprising secondary term, which is similar to (but subtly different from) the secondary term in the counting function of real cubic fields. We include technical details of our computation method, which uses a modified infrastructure approach implemented on GPUs.

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