New Zealand Journal of Mathematics

On the Waring problem with Dickson polynomials modulo a prime

Igor E. Shparlinski, José Felipe Voloch — Tue, 15 Jul 2025 00:00:00 +1200

We improve recent results of D. Gomez and A. Winterhof (2010) and of A. Ostafe and I. E. Shparlinski (2011) on the Waring problem with Dickson polynomials in the case of prime finite fields. Our approach is based on recent bounds of Kloosterman and Gauss sums due to A. Ostafe, I. E. Shparlinski and J. F. Voloch (2021).

An esoteric identity with many parameters and other elliptic extensions of elementary identities

Gaurav Bhatnagar, Archna Kumari, Michael Schlosser — Tue, 15 Jul 2025 00:00:00 +1200

We provide elliptic extensions of elementary identities such as the sum of the first $n$ odd or even numbers, the geometric sum and the sum of the first $n$ cubes. Many such identities, and their $q$-analogues, are indefinite sums, and can be obtained from telescoping. So we used telescoping in our study to find elliptic extensions of these identities. In the course of our study, we obtained an identity with many parameters, which appears to be new even in the $q$-case.
In addition, we recover some $q$-identities due to Warnaar.

New parameterized mock theta functions and Hecke-type double sums

Li-Jun Hao, Liang-Liang Xu — Tue, 15 Jul 2025 00:00:00 +1200

In this paper we express two new mock theta functions with one parameter as the Appell-Lerch sums using the Bailey mechanism. Meanwhile, we also obtain some Hecke-type double sums for some new $q$-series. In addition, we establish the relationships between the mock theta functions and the classical sixth and eighth order mock theta functions.
Furthermore, we give the Hecke-type double sums for the second order mock theta function $D_5(q)$.

On the Cauchy integral theorem and Polish spaces

Cristian López Morales, Camilo Ramírez Maluendas — Thu, 14 Aug 2025 00:00:00 +1200

We prove that if a continuous function $f$ in an open subset $U\subset\mathbb{C}$ is analytic in $U\setminus X$, where $X\subset U$ is a Polish space having characteristic system $(k,n)\in\mathbb N_0\times\mathbb N$, then the complex line integral of $f$ along the boundary of any triangle in $U$ vanishes.

Classification of connected shelves

Mohamed Elhamdadi, Neranga Fernando, Mathew Goonewardena — Tue, 09 Sep 2025 00:00:00 +1200

We investigate finite right-distributive binary algebraic structures called shelves. We first use symbolic computations with Python to classify (up to isomorphism) all connected shelves with order less than six. We explore the group structure generated by the rows of latin shelves. We also define two-variable shelf polynomial by analogy with the quandle polynomial and then state a conjecture about connected idempotent shelves.

An inverse problem for renormalized area: determining the bulk metric with minimal surfaces

Jared Marx-Kuo — Wed, 10 Sep 2025 00:00:00 +1200

We present an inverse problem which uses the renormalized area functional on minimal submanifolds to determine the asymptotic expansion of asymptotically hyperbolic, conformally compact metrics which are partially even to high order. We use a rigidity argument to determine the conformal infinity of the metric via the renormalized area. We then consider the renormalized volume of perturbations of the hemisphere to determine the higher order terms in the asymptotic expansion of the metric. We prove global rigidity when these metrics are log-analytic, and further note that renormalized area determines the obstruction tensor for PE metrics.