On this site we list the polynomials of a (degrevlex) Gröbner basis of relations among eta functions for various levels.
The relations are expressed as polynomials in variables E1, E2, etc. where for a divisor δ of a given level N the variable Eδ stands for the eta function η(δτ).
η(δτ) = exp(πiτ/12) Product(1 − q^(δn), n=1..infinity) with q = q(τ) = exp(2πiτ)
The relations listed by Somos can easily be expressed in terms of the Eδ variables (i.e. directly in terms of eta functions). These (transformed) relations can be represented in terms of the Gröbner bases from this site. We give a collection of such representations for various entries in the collection of Somos.
The relations have been computed by an implentation of the
algorithm samba from Dancing Samba
with Ramanujan Partition Congruences in the computer algebra
system FriCAS and the
slimgb (Gröbner bases) algorithm from the system Singular.
All files can be read directly by FriCAS, in particular,
fricassomos*.input files evaluates the
representation of relations from Somos' list in terms of the
Gröbner basis of relations and thus yields a list of zeros. See
for an example of evaluation in FriCAS
for a FriCAS implementation to translate the notation of Somos
into our notation.
Details of how exactly the relations have been computed is explained in the article Construction of all Polynomial Relations among Dedekind Eta Functions of Level N by Ralf Hemmecke and Silviu Radu in the Journal of Symbolic Computation. The article is also available as RISC report 18-03.