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Here we give an algorithm to determine Shintani domains for non-totally complex number fields from the signed fundamental domains given in the works of Diaz y Diaz, Espinoza and Friedman. We also present some explicit examples of Shintani domains.

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Computing Shintani fundamental domains

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:

COMPUTING SHINTANI DOMAINS

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

FILE DESCRIPTION:

  1. 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
  2. Using SignedDomain.gp, we give in the file ShintaniDomain.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:

$$F:=[F_1,F_2,F_3];$$

The first entry $F_1$ (i.e., $F[1]$) has the form

$$[t, p, reg, disc, [r_1, r_2], U, T]$$

with

$t =$ real computation time for $F$ in milliseconds (this depends on the number of negative cones in a signed domain and also on the type of processor used)

$p =$ irreducible polynomial defining a non-totally complex number field $k := \text{the quotient ring }\mathbb{Q}[X]/(p)$ of degree $n$

$reg =$ Regulator of $k$ to $19$ decimals

$disc =$ Discriminant of $k$

$[r_1, r_2]=[\text{Number of real places}, \text{Number of complex places}]$ signature of $k$, so $n=r_1+2r_2$

$U =$ fundamental units of $k$. The Shintani domain corresponds to the action on $(\mathbb{R}_{+})^{r_1}\times(\mathbb{C}^{\ast})^{r_2}$ of the group generated by $U$ (for $r_1>0$ and rank of units $r=r_1+r_2-1>0$). The units in $U$, like all other elements of $k$, is given as a polynomial $g$ in $\mathbb{Q}[X]$ of degree at most $n$. The associated element of $k$ is the class of $g$ in $\mathbb{Q}[X]/(p)$

$T =$ number of semi-closed $n$-dimensional cones in the Shintani domain constructed.

The second entry $F_2$ (i.e., $F[2]$) has the form

$$[C_1,C_2,...,C_T]$$

which is a list of the $T$ (semi-closed n-dimensional) cones in the Shintani domain. Here $T=F[1][7]$ is the last entry of $F_1$ described above. Each cone $C_j$ is given by $m$ linear inequalities ($m$ depending on the cone) giving $m$ closed or open half-spaces whose intersection is $C_j$. Thus, each $C_j$ has the form

$$[v_1,v_2,...,v_m]$$

where $v_i=[w,1]$ or $[w,-1]$ and $w$ is an element of $k$ (depending on $i$ and $j$). If $w$ is followed by $1$, then the corresponding (closed) half-space is the set of elements $x$ of $\mathbb{R}^n$ with $\text{Trace}(xw)\geq 0$. If $w$ is followed by $-1$, then the corresponding (open) half-space is given by $\text{Trace}(xw)>0$. Here Trace is the extension to $\mathbb{R}^n$ of the trace map from $k$ to $\mathbb{Q}$.

The third entry $F_3$ of $F$ (i.e., $F[3]$) has the form

$$[CC_1,CC_2,...,CC_T]$$

where $CC_j$ is the closure in $\mathbb{R}^n$ of the cone $C_j$ in $F_2$. Each closed cone $CC_j$ is given here by a list of generators in $k$.

  1. 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

$S[2,1]:=$ShintaniK21

$S[3,1]:=$ShintaniK31

$S[3,2]:=$ShintaniK32

$S[4,2]:=$ShintaniK42

$S[4,3]:=$ShintaniK43

$S[5,2]:=$ShintaniK52

$S[5,3]:=$ShintaniK53

$S[5,4]:=$ShintaniK54

$S[6,5]:=$ShintaniK65

Each of these folders $S[n,r]$ respectively contain fundamental domains of some number fields of degree $n$ for $n=2,3,4,5,6$ with rank of units $r=r_1+r_2-1$ for $r=1,2,3,4,5$ such that $r_1>0$.

Each folder $S[n,r]$ contains three files, which we denote here as:fieldsKnr.gp, ShintaniKnr.txt and ShintaniKnr-ML.sage (Note the suffix $nr$ in the names of these files, where $nr$ means $[n,r]$. For example, if $[n,r]=[5,3]$ then the folder $S[5,3]$ contains the files: fields53.gp, ShintaniK53.txt and ShintaniK53-ML.sage)

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 command

    load('ShintaniKnr-ML.sage')

this returns the same list of examples as the file ShintaniKnr.txt with the same structure.

SOME REMARKS:

(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 $n$ (using $r_1>0$ and rank $r=r_1+r_2-1>0$).

(2) The fundamental domains in the folder $S[2,1]$ correspond to totally real quadratic fields which are widely known by number theorists. See for example Borevich-Shafarevich's Book "Number theory" (Chapter 5, Section 1.2).

(3) And the folder $S[3,2]$ which correspond to Shintani domains for totally real cubic fields are also known, see for example Diaz y Diaz and Friedman's work: Real Cubic Shintani

(4) On the other hand, the folder $S[3,1]$ contains Shintani domains for complex cubic number fields, this is consistent with a recently published work: Complex Cubic Shintani

(5) The remaining files: $S[4,2]$, $S[4,3]$, $S[5,2]$, $S[5,3]$, $S[5,4]$ and $S[6,5]$ contains Shintani domains (for a list of polynomials given) of non-totally complex quartic, quintic and sextic fields.

(6) The main bottleneck is that the number of cones on a Signed Domain grows (this is $3^{r_2}(n-1)!$ cones) when $n$ grows. This number of cones is the input in our main algorithm to obtain a Shintani domain, so our implementation works well when $n\leq 5$ and sometimes in sextic fields $n=6$ when the negative cones in a Signed domains is small. However, it is possible trying to compute Shintani domains of (non-totally complex) number fields for $degree>6$ if you are using a good processor.

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Here we give an algorithm to determine Shintani domains for non-totally complex number fields from the signed fundamental domains given in the works of Diaz y Diaz, Espinoza and Friedman. We also present some explicit examples of Shintani domains.

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