Construction of all polynomial relations among Dedekind eta functions of level \(N\)
type: Item
instance of: scholarly article
title: Construction of all polynomial relations among Dedekind eta functions of level \(N\)
zbMATH Open document ID: 1430.13046
author: Ralf Hemmecke
author: Silviu Radu
published in: Journal of Symbolic Computation
publication date: 2019-06-28T00:00:00Z
review text: The aim of this paper is to provide an algorithm to compute a Gröbner basis for the relations among Dedekind \(\eta\)-functions. Let \(\mathbb{H} = \{c\in\mathbb{C} \mid \Im(c)> 0 \}\) denote the complex upper half-plane and \[ \eta: \mathbb{H} \to \mathbb{C},\ \eta(\tau)=\exp(\pi i\tau/12)\prod_{k=1}^{\infty}(1-\exp(2\pi ik \tau))) \] Let us fix a positive integer \(N\) and let \(1 = \delta_1 < \delta_2 < \dots < \delta_n = N\) be the positive divisors of \(N\). We define \(\phi : \mathbb{Q}[E_{\delta_1} , \dots , E_{\delta_n}]\to \mathbb{Q}[\eta(\delta_1 \tau),\dots,\eta(\delta_n \tau)]\) with \(\phi(E_i)=\eta(\delta_i \tau)\) for each \(i\) where \(E_{\delta_1} , \dots , E_{\delta_n}\) is a sequence of variables. The main contribution of the paper is to compute a Gröbner basis for the kernel of \(\phi\).
reviewed by: Amir Hashemi
Mathematics Subject Classification ID: 13P10
Mathematics Subject Classification ID: 68W30
zbMATH DE Number: 7074700
zbMATH Keywords: Dedekind \(\eta\) function
zbMATH Keywords: modular functions
zbMATH Keywords: modular equations
zbMATH Keywords: ideal of relations
zbMATH Keywords: Gröbner basis
describes a project that uses: QEta
describes a project that uses: SINGULAR
describes a project that uses: 4ti2
describes a project that uses: SageMath
describes a project that uses: Dependencies
MaRDI profile type: Publication
full work available at URL: j.jsc.2018.10.001
OpenAlex ID: W2895333579
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Wikidata QID: Q129126414
DOI: 10.1016/J.JSC.2018.10.001
description: scientific article; zbMATH DE number 7074700