Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory pp 259-268 | Cite as
The Generators of all Polynomial Relations Among Jacobi Theta Functions
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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 \(K :=\mathbb {Q}(\theta _2(0|\tau ),\theta _3(0|\tau ),\theta _4(0|\tau ))\) is generated by just two polynomials, that correspond to well known identities among Jacobi theta functions.
References
- 1.F.W.J. Olver, A.B. Olde Daalhuis, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders (eds.), NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/. Release 1.0.19 of 2018-06-22
- 2.L. Ye, Elliptic function based algorithms to prove Jacobi theta function relations. J. Symb. Comput. 89, 171–193 (2018)MathSciNetCrossRefGoogle Scholar
- 3.E.T. Whittaker, G.N. Watson, A Course of Modern Analysis, 4th edn. (Cambridge University, Cambridge, 1927). Reprinted 1965Google Scholar
- 4.Sims C.C, Computation with Finitely Presented Groups. Encyclopedia of Mathematics and Its Applications, vol. 48 (Cambridge University, Cambridge, 1994)Google Scholar
- 5.B. Buchberger, Ein Algorithmus zum Auffinden der Basiselemente des Restklassenringes nach einem nulldimensionalen Polynomideal. Ph.D. thesis, University of Innsbruck, Department of Mathematics, Innsbruck, Austria (1965)Google Scholar
- 6.T. Becker, V. Weispfenning (in cooperation with H. Kredel), Gröbner Bases. A Computational Approach to Commutative Algebra. Graduate Texts in Mathematics, vol. 141 (Springer, New York, 1993)CrossRefGoogle Scholar
- 7.R. Hemmecke, S. Radu, Construction of all polynomial relations among Dedekind eta functions of level \({N}\), RISC Report Series 18-03 Research Institute for Symbolic Computation, Johannes Kepler Universität, Linz/A (2018)Google Scholar
- 8.B.C. Berndt, Ramanujan’s Notebooks (Springer, Berlin, 1997)Google Scholar
- 9.FriCAS team, Fricas—an advanced computer algebra system. Available at http://fricas.sf.net
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