cohn3 -- modular equations for special algebraic number fields

Synopsis

Usage:

cohn3(kk)
Inputs:
- kk, a ring, the coefficient ring
Outputs:
- a list, of the polynomials in the system

Description

This system was solved in May 2020, using solveSystem in Macaulay2 v1.15 with an Intel(R) Core(TM) i5-4258U CPU at 2.40GHz.

There were 72 solutions found in 13.8985 seconds (with a Bezout bound of 1080).

Reference: See the PoSSo test suite. Andre' Galligo and Carlo Traverso. "Practical Determination of the dimension of an algebraic variety", in E. Kaltofen and S.M. Watt, Eds "Computers and Mathematics", pages 46-52, 1989.

H. Cohn. "An explicit modular equation in two variables and Hilbert's Twelfth problem", Math. of Comp. 38, pp. 227-236, 1982.

H. Cohn, J. Deutch. "An explicit modular equation in two variables for Q[sqrt(3)]", Math. of Comp. 50, pp. 557-568, 1988.

See also: http://homepages.math.uic.edu/~jan/Demo/cohn3.html

i1 : F = cohn3(QQ)

         3 2     2 2     2   2       2 2       2            2           2  
o1 = {- x y  + 2x y z - x y*z  - 144x y  - 207x y*z + 288x*y z + 78x*y*z  +
     ------------------------------------------------------------------------
        3        2           2                     2                         
     x*z  - 3456x y - 5184x*y  - 9504x*y*z - 432x*z  - 248832x*y + 62208x*z -
     ------------------------------------------------------------------------
                  3   2    2 2 2     3        2 2       3 2      2   2  
     2985984x, - x z*t  + x z t  - 6x z*t + 4x z t + 32x t  - 72x z*t  -
     ------------------------------------------------------------------------
          2 2    3 2     3        2            2             2       2 2  
     87x*z t  - z t  - 8x z - 432x z*t - 414x*z t + 2592x*z*t  + 864z t  -
     ------------------------------------------------------------------------
          2                      2             2                           
     1728x z - 20736x*z*t + 3456z t - 186624z*t  - 124416x*z - 1492992z*t -
     ------------------------------------------------------------------------
                2   3       2 3    3 3     2   2        2 2     3 2  
     2985984z, x y*t  - 2x*y t  + y t  + 8x y*t  - 12x*y t  + 4y t  -
     ------------------------------------------------------------------------
            3      2 3      2           2            2      2 2         3  
     24x*y*t  + 24y t  + 20x y*t - 20x*y t - 160x*y*t  + 96y t  + 128x*t  +
     ------------------------------------------------------------------------
        2                     2                                 3 3    2   3
     16x y + 96x*y*t + 2304x*t  + 1152x*y + 13824x*t + 27648x, y t  - y z*t 
     ------------------------------------------------------------------------
         3 2     2   2      2 3          3       2 2           2     2 2  
     + 4y t  - 2y z*t  + 72y t  + 71y*z*t  + 288y t  + 360y*z*t  + 6z t  +
     ------------------------------------------------------------------------
            3         3                2           2          2         3  
     1728y*t  - 464z*t  + 432y*z*t + 8z t + 6912y*t  - 4320z*t  + 13824t  +
     ------------------------------------------------------------------------
      2                    2
     z  - 13824z*t + 55296t  - 13824z}

o1 : List

Ways to use cohn3 :

cohn3(Ring)

For the programmer

The object cohn3 is a method function.