Here we give an algorithm (in Pari/GP) to obtain a TRUE fundamental domain from a SIGNED fundamental domain for the action of the totally positive units group of a NON-TOTALLY COMPLEX NUMBER FIELD. We also present some examples of Shintani domains in the folder Examples. This implementation is based in the manuscript:
by A. CAPUÑAY, International Journal of Number Theory, Vol. 20, No. 02, pp. 393-411 (2024).
The SIGNED domains were established in the works of Diaz y Diaz, Espinoza and Friedman:
[DDF14] F. Diaz y Diaz and E. Friedman, "Signed fundamental domain for totally real number fields" (2014)
MR4105945
[EF20] M. Espinoza and E. Friedman, "Twisters and Signed fundamental domains of number fields" (2020)
MR3198753
Our implementation is also based in the description of rational cones by inequalities (or H-representation) and
generators (or V-representation). For this we use the Fukuda-Prodon's paper:
[FP96] Fukuda and Prodon, "Double description method revisited" (1996)
MR1448924
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In the file
SignedDomain.gp
we implement the signed domains given in [DDF14] (for totally real fields) and [EF20] (for non-totally complex fields). Which can be read in Pari/GP using the command\r SignedDomain.gp
-
Using
SignedDomain.gp
, we give in the fileShintaniDomain.gp
our main algorithm to find a true fundamental domain from a signed one. This can be read using\r ShintaniDomain.gp
So given as input an irreducible polynomial p
(which defines a non-totally complex number field), the command in Pari/GP
F=fudom(p);
return a Shintani fundamental domain with the following structure:
The first entry
with
The second entry
which is a list of the
where
The third entry
where
- In the folder Examples we show several examples of explicit Shintani domains obtained using the
fudom(p)
command described in item 2. Here there exists 9 subfolders
Each of these folders
Each folder fieldsKnr.gp
, ShintaniKnr.txt
and ShintaniKnr-ML.sage
(Note the suffix
Where:
-
The file
fieldsKnr.gp
contains a data of fields used to obtains Shintani domains which was download from https://www.lmfdb.org/ -
The file
ShintaniKnr.txt
contains a data of explicit Shitani domains which can be read by Pari/GP using the command\r ShintaniKnr.txt
This returns a vector called
$examples=[E_1,E_2,...,E_g]$ , where each$E_i=fudom(p)$ is a vector of size three which was described in item 2 with$p$ an irreducible polynomial of degree$n$ which defines a non-totally complex number field$k$ with rank of units$r$ . -
The file
ShintaniKnr-ML.sage
can be read by SageMath using the commandload('ShintaniKnr-ML.sage')
this returns the same list of examples as the file ShintaniKnr.txt
with the same structure.
(1) After uploading files SignedDomain.gp
and ShintaniDomain.gp
, the command (Pari/GP)
ShintExamples(L)
returns a file with a list of examples of the calculated Shintani domains, where L=vector of irreducible polynomials of degree
(2) The fundamental domains in the folder
(3) And the folder
(4) On the other hand, the folder
(5) The remaining files:
(6) The main bottleneck is that the number of cones on a Signed Domain grows (this is