README.txt

   This README file is meant to aid in interpreting the accompanying files SHINTANI.txt and SHINTANI-ML.sage, which gives examples attached to particular number fields k of the algorithm described in section 6 of the manuscript (abbreviated MS)

    ATTRACTOR-REPELLER CONSTRUCTION OF SHINTANI DOMAINS FOR 
    TOTALLY COMPLEX QUARTIC FIELDS

by A. CAPUÑAY, M. ESPINOZA AND E. FRIEDMAN, Journal of Number Theory (2024).
https://www.sciencedirect.com/science/article/pii/S0022314X23002299
 
   The file SHINTANI.txt is a text file meant to be downloaded and read into PARI GP via the command

        \r SHINTANI.txt

It can, of course, be read and used by other software, bearing in mind that square brackets [ ] are used for lists, its elements are separated by commas. The file SHINTANI-ML.sage can be read by SageMath using 

        load('SHINTANI-ML.sage')
             
This returns the same list of examples as the file SHINTANI.txt and has the same structure.

The file SHINTANI.txt contains 11 examples of Shintani domains E1,...,E11. It consists of one long line of text of the form 

examples = [E1,E2,E3,E4,E5,E6,E7,E8,E9,E10,E11];

ending in a semicolon (to prevent PARI GP from printing this big  file to screen). Each Ei (i.e., examples[i] for 1 <= i <= 11) has the form  

              [a1,a2,a3,a4]

The first entry a1 (i.e., examples[i][1]) has the form 

            [t,p,reg,disc,E,r,T]

with 

t    = real computation time for Ei in milliseconds
p    = quartic irreducible polynomial defining a totally 
       complex number field k := the quotient ring Q[X]/(p)  
reg  = Regulator of k to 19 decimals
disc = Discriminant of k
E    = fundamental unit of k. The Shintani domain corresponds to
       the action on C^* x C^* of the group generated by E. The 
       unit E, like all other elements of k below, is given as a 
       polynomial g in Q[X] of degree at most 3. The associated 
       element of k is the class of g in Q[X]/(p)
r    = minimal positive integer with |E_1|^{2r} < 0.184 as in  
       display (34) of Ms. Here E_1 is an embedding of E in C     
T    = number of 4-cones in the Shintani domain constructed 

   The second entry a2 of Ei (i.e., examples[i][2]) has the form  

              [ee1,ee2,ee3,ee4] 

where eej (for 1 <= j <= 4) is an element of k approximating the j-th element of the standard basis of R^4 = C^2, wherein k is embedded. The elements eej are denoted (in the Latex source code) $\tilde{e}_j$ at the beginning of the proof of the Main Theorem in section 5 of the Ms. The error bound ($\varepsilon$ in the Ms) used was 1/150. 

    The third entry a3 of Ei (i.e., examples[i][3]) has the form  

              [C1,C2,...,CT]

which is a list of the T (semi-closed) cones in the Shintani domain. Here T = examples[i][1][7] is the last entry of a1  described above. Each cone Cj is given by m linear inequalities (m depending on the cone) giving m closed or open half-spaces whose intersection is Cj. Thus, each Cj has the form  

              [v1,v2,...,vm]

where vi=[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 R^4 with Trace(xw) >= 0. If w is followed by -1, then the corresponding (open) half-space is given by Trace(xw) > 0. Here Trace is the extension to R^4 of the trace map from k to Q.

   The fourth entry a4 of Ei (i.e., examples[i][4]) has the form  

              [CC1,CC2,...,CCT]

where CCj is the closure in R^4 of the cone Cj in a3. Each closed cone CCj is given here by a list of generators in k.