The Macaulay session which supports Example 6 in Lecture 1 on 
``Socle degrees of Frobenius powers''.

This Macaulay session calculates the socle degrees of R/J^{[p^e]} for 
for R=ZZ/2[x,y,z]/(f), where f= x^5+y^5+z^5 and J=(x,y,z).

We learn:

e soc degrees
0 0:1
1 3:1
2 9:1
3 12:1 16:1
4 22:1 30:1
5 42:1 58:1



kustin@berry 8 % !M
M2
Macaulay 2, version 0.9.2
--Copyright 1993-2001, D. R. Grayson and M. E. Stillman
--Singular-Factory 2.0.5, copyright 1993-2001, G.-M. Greuel, et al.
--Singular-Libfac 2.0.4, copyright 1996-2001, M. Messollen
 
i1 : P=ZZ/2[x,y,z];
 
i2 : J=ideal(x , y, z);
 
o2 : Ideal of P
 
i3 : FrobPower = (K,e) -> (
p:=char ring K;
ans:=ideal(K_0^(p^e));
count:=1;
while count < numgens( K ) do(
ans=ans+ideal(K_count^(p^e));
count=count+1);
return(ans));
                                                      
i4 : f=x^5+y^5+z^5;
 
i5 : betti basis( ((FrobPower(J,0)+ideal(f)):ideal (x,y,z))/((FrobPower(J,0)+ideal(f))))
 
o5 = total: 1 1
        -1: . 1
         0: 1 .
 
i6 : betti basis( ((FrobPower(J,1)+ideal(f)):ideal (x,y,z))/((FrobPower(J,1)+ideal(f))))
 
o6 = total: 4 1
         2: 3 1
         3: 1 .
 
i7 : betti basis( ((FrobPower(J,2)+ideal(f)):ideal (x,y,z))/((FrobPower(J,2)+ideal(f))))
 
o7 = total: 4 1
         4: 3 .
         5: . .
         6: . .
         7: . .
         8: . 1
         9: 1 .
 
i8 :
     betti basis( ((FrobPower(J,4)+ideal(f)):ideal (x,y,z))/((FrobPower(J,4)+ideal(f))))
 
o8 = total: 6 2
         5: 1 .
         6: . .
         7: . .
         8: . .
         9: . .
        10: . .
        11: . .
        12: . .
        13: . .
        14: . .
        15: . .
        16: 3 .
        17: . .
        18: . .
        19: . .
        20: . .
        21: . 1
        22: 1 .
        23: . .
        24: . .
        25: . .
        26: . .
        27: . .
        28: . .
        29: . 1
        30: 1 .
 
i9 : betti basis( ((FrobPower(J,3)+ideal(f)):ideal (x,y,z))/((FrobPower(J,3)+ideal(f))))
 
o9 = total: 6 2
         5: 1 .
         6: . .
         7: . .
         8: 3 .
         9: . .
        10: . .
        11: . 1
        12: 1 .
        13: . .
        14: . .
        15: . 1
        16: 1 .
 
i10 : betti basis( ((FrobPower(J,5)+ideal(f)):ideal (x,y,z))/((FrobPower(J,5)+ideal(f))))
 
o10 = total: 6 2
          5: 1 .
          6: . .
          7: . .
          8: . .
          9: . .
         10: . .
         11: . .
         12: . .
         13: . .
         14: . .
         15: . .
         16: . .
         17: . .
         18: . .
         19: . .
         20: . .
         21: . .
         22: . .
         23: . .
         24: . .
         25: . .
         26: . .
         27: . .
         28: . .
         29: . .
         30: . .
         31: . .
         32: 3 .
         33: . .
         34: . .
         35: . .
         36: . .
         37: . .
         38: . .
         39: . .
         40: . .
         41: . 1
         42: 1 .
         43: . .
         44: . .
         45: . .
         46: . .
         47: . .
         48: . .
         49: . .
         50: . .
         51: . .
         52: . .
         53: . .
         54: . .
         55: . .
         56: . .
         57: . 1
         58: 1 .
 
i11 :