
R = frac(QQ[x]/(x^5-2)[y])
F = (x/(y-2) + 1 + x*y)
G = F^2*F

f = numerator F^3
g = denominator F^3
f/g
f^3
g^3
(f^3) // (g^3)
S = frac(QQ[x,y]/(x^5-2))
syz matrix{{denominator G, numerator G}}

i16 : use frac(QQ[x]/(x^5-2)[y])

          //QQ [x]\   \
o16 = frac||------|[y]|
          || 5    |   |
          \\x  - 2/   /

o16 : FractionField

i17 : (x/(y-2) + 1 + x*y)^4

                                       4 8                                   
      54793191022000727827015529616000x y  + (- 43834552817600582261612423692
o17 = -----------------------------------------------------------------------
                                                                             
                                                                             
      -----------------------------------------------------------------------
           4                                     3  7                        
      8000x  + 219172764088002911308062118464000x )y  + (15342093486160203791
      -----------------------------------------------------------------------
                                                                             
                                                                             
      -----------------------------------------------------------------------
                     4                                      3                
      56434829248000x  - 1753382112704023290464496947712000x  + 3287591461320
      -----------------------------------------------------------------------
                                                                             
                                                                             
      -----------------------------------------------------------------------
                           2  6                                         4    
      04366962093177696000x )y  + (- 3068418697232040758312869658496000x  + 5
      -----------------------------------------------------------------------
                                                                             
                                                                             
      -----------------------------------------------------------------------
                                        3                                    
      917664630376078605317677198528000x  - 263007316905603493569674542156800
      -----------------------------------------------------------------------
                                                                             
                                                                             
      -----------------------------------------------------------------------
        2                                       5                            
      0x  + 219172764088002911308062118464000x)y  + (383552337154005094789108
      -----------------------------------------------------------------------
                                                                             
                                                                             
      -----------------------------------------------------------------------
                 4                                       3                   
      7073120000x  - 10958638204400145565403105923200000x  + 8547737799432113
      -----------------------------------------------------------------------
                                                                             
                                                                             
      -----------------------------------------------------------------------
                         2                                                   
      541014422620096000x  - 1753382112704023290464496947712000x + 5479319102
      -----------------------------------------------------------------------
                                                                             
                                                                             
      -----------------------------------------------------------------------
                              4                                         4    
      2000727827015529616000)y  + (- 3068418697232040758312869658496000x  + 1
      -----------------------------------------------------------------------
                                                                 4           
                                54793191022000727827015529616000y  - 43834552
      -----------------------------------------------------------------------
                                         3                                   
      2054502024840160121943416515520000x  - 14465402429808192146332099818624
      -----------------------------------------------------------------------
                                3                                      2
      8176005822616124236928000y  + 1315036584528017467848372710784000y  - 17
      -----------------------------------------------------------------------
          2                                                                  
      000x  + 5479319102200072782701552961600000x - 4383455281760058226161242
      -----------------------------------------------------------------------

      53382112704023290464496947712000y + 876691056352011645232248473856000
      -----------------------------------------------------------------------
                3                                       4                    
      36928000)y  + (1534209348616020379156434829248000x  - 78902195071681048
      -----------------------------------------------------------------------
      -----------------------------------------------------------------------
                        3                                       2            
      07090236264704000x  + 13479124991412179045445820285536000x  - 832856503
      -----------------------------------------------------------------------
      -----------------------------------------------------------------------
                                                                       2     
      5344110629706360501632000x + 1315036584528017467848372710784000)y  + (-
      -----------------------------------------------------------------------
      -----------------------------------------------------------------------
                                         4                                   
       438345528176005822616124236928000x  + 28492459331440378470048075400320
      -----------------------------------------------------------------------
      -----------------------------------------------------------------------
         3                                      2                            
      00x  - 6575182922640087339241863553920000x  + 6136837394464081516625739
      -----------------------------------------------------------------------
      -----------------------------------------------------------------------
                                                                             
      316992000x - 1753382112704023290464496947712000)y + 5479319102200072782
      -----------------------------------------------------------------------
      -----------------------------------------------------------------------
                    4                                     3                  
      7015529616000x  - 438345528176005822616124236928000x  + 131503658452801
      -----------------------------------------------------------------------
      -----------------------------------------------------------------------
                          2
      7467848372710784000x  - 1753382112704023290464496947712000x + 876691056
      -----------------------------------------------------------------------
      -----------------------------------------------------------------------
      352011645232248473856000
      ------------------------

          //QQ [x]\   \
o17 : frac||------|[y]|
          || 5    |   |
          \\x  - 2/   /

