The Generators of all Polynomial Relations among Jacobi Theta Functions

Ralf Hemmecke; Silviu Radu; Liangjie Ye

doi:10.1007/978-3-030-04480-0_11

The Generators of all Polynomial Relations among Jacobi Theta Functions

Ralf Hemmecke, Silviu Radu, Liangjie Ye

Research Institute for Symbolic Computation

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

Abstract

In this article, we consider the classical Jacobi theta functions $\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomial relations among them with coefficients in $K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ is generated by just two polynomials, that correspond to well known identities among Jacobi theta functions. Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf

Original language	English
Title of host publication	Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory
Editors	Johannes Blümlein and Carsten Schneider and Peter Paule
Place of Publication	Cham
Publisher	Springer International Publishing
Pages	259-268
Number of pages	9
ISBN (Print)	978-3-030-04479-4
DOIs	https://doi.org/10.1007/978-3-030-04480-0_11
Publication status	Published - 2019

Publication series

Name	Texts & Monographs in Symbolic Computation

Fields of science

101 Mathematics
101001 Algebra
101005 Computer algebra
101009 Geometry
101012 Combinatorics
101013 Mathematical logic
101020 Technical mathematics

JKU Focus areas

Digital Transformation

Access to Document

10.1007/978-3-030-04480-0_11

Cite this

Hemmecke, R., Radu, S., & Ye, L. (2019). The Generators of all Polynomial Relations among Jacobi Theta Functions. In Johannes Blümlein and Carsten Schneider and Peter Paule (Ed.), Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory (pp. 259-268). (Texts & Monographs in Symbolic Computation). Springer International Publishing. https://doi.org/10.1007/978-3-030-04480-0_11

Hemmecke, Ralf ; Radu, Silviu ; Ye, Liangjie. / The Generators of all Polynomial Relations among Jacobi Theta Functions. Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. editor / Johannes Blümlein and Carsten Schneider and Peter Paule. Cham : Springer International Publishing, 2019. pp. 259-268 (Texts & Monographs in Symbolic Computation).

@inbook{29bfcee1d002475e86aedae4a24dff31,

title = "The Generators of all Polynomial Relations among Jacobi Theta Functions",

abstract = "In this article, we consider the classical Jacobi theta functions $\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomial relations among them with coefficients in $K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ is generated by just two polynomials, that correspond to well known identities among Jacobi theta functions. Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf",

author = "Ralf Hemmecke and Silviu Radu and Liangjie Ye",

year = "2019",

doi = "10.1007/978-3-030-04480-0_11",

language = "English",

isbn = "978-3-030-04479-4",

series = "Texts & Monographs in Symbolic Computation",

publisher = "Springer International Publishing",

pages = "259--268",

editor = "{Johannes Bl{\"u}mlein and Carsten Schneider and Peter Paule}",

booktitle = "Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory",

}

Hemmecke, R , Radu, S & Ye, L 2019, The Generators of all Polynomial Relations among Jacobi Theta Functions. in Johannes Blümlein and Carsten Schneider and Peter Paule (ed.), Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Texts & Monographs in Symbolic Computation, Springer International Publishing, Cham, pp. 259-268. https://doi.org/10.1007/978-3-030-04480-0_11

The Generators of all Polynomial Relations among Jacobi Theta Functions. / Hemmecke, Ralf ; Radu, Silviu; Ye, Liangjie.
Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. ed. / Johannes Blümlein and Carsten Schneider and Peter Paule. Cham: Springer International Publishing, 2019. p. 259-268 (Texts & Monographs in Symbolic Computation).

Research output: Chapter in Book/Report/Conference proceeding › Chapter › peer-review

TY - CHAP

T1 - The Generators of all Polynomial Relations among Jacobi Theta Functions

AU - Hemmecke, Ralf

AU - Radu, Silviu

AU - Ye, Liangjie

PY - 2019

Y1 - 2019

N2 - In this article, we consider the classical Jacobi theta functions $\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomial relations among them with coefficients in $K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ is generated by just two polynomials, that correspond to well known identities among Jacobi theta functions. Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf

AB - In this article, we consider the classical Jacobi theta functions $\theta_i(z)$, $i=1,2,3,4$ and show that the ideal of all polynomial relations among them with coefficients in $K :=\setQ(\theta_2(0|\tau),\theta_3(0|\tau),\theta_4(0|\tau))$ is generated by just two polynomials, that correspond to well known identities among Jacobi theta functions. Also available as RISC Report 18-09 http://www.risc.jku.at/publications/download/risc_5719/thetarelations.pdf

U2 - 10.1007/978-3-030-04480-0_11

DO - 10.1007/978-3-030-04480-0_11

M3 - Chapter

SN - 978-3-030-04479-4

T3 - Texts & Monographs in Symbolic Computation

SP - 259

EP - 268

BT - Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

A2 - Johannes Blümlein and Carsten Schneider and Peter Paule, null

PB - Springer International Publishing

CY - Cham

ER -

Hemmecke R , Radu S, Ye L. The Generators of all Polynomial Relations among Jacobi Theta Functions. In Johannes Blümlein and Carsten Schneider and Peter Paule, editor, Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory. Cham: Springer International Publishing. 2019. p. 259-268. (Texts & Monographs in Symbolic Computation). doi: 10.1007/978-3-030-04480-0_11