@**techreport**{RISC5561,author = {Ralf Hemmecke and Silviu Radu},

title = {{Construction of all Polynomial Relations among Dedekind Eta Functions of Level $N$}},

language = {english},

abstract = {We describe an algorithm that, given a positive integer $N$,
computes a Gr\"obner basis of the ideal of polynomial relations among Dedekind
$\eta$-functions of level $N$, i.e., among the elements of
$\{\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)\}$ where
$1=\delta_1<\delta_2\dots<\delta_n=N$ are the positive divisors of
$N$.
More precisely, we find a finite generating set (which is also a
Gr\"obner basis of the ideal $\ker\phi$ where
\begin{gather*}
\phi:Q[E_1,\ldots,E_n] \to Q[\eta(\delta_1\tau),\ldots,\eta(\delta_n\tau)],
\quad
E_k\mapsto \eta(\delta_k\tau),
\quad
k=1,\ldots,n.
\end{gather*}
},

number = {18-03},

year = {2018},

month = {January 26},

keywords = {Dedekind $\eta$ function, modular functions, modular equations, ideal of relations, Groebner basis},

length = {16},

type = {RISC Report Series},

institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},

address = {Schloss Hagenberg, 4232 Hagenberg, Austria}

}