The Macaulay session which supports Example 5 in Lecture 1 on 
``Socle degrees of Frobenius powers''.

This Macaulay session calculates the socle degrees of R/J^{[p^e]} for 
R= ZZ/2[x,y,z]/(f), where f= x^3+y^3+z^3 and J= (x,y,z).

We learn:

e socle degrees
0 0:1
1 3:1
2 6:2
3 12:2
4 24:2

kustin@berry 7 % !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^3+y^3+z^3;
 
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: 6 2
         3: 1 .
         4: 3 .
         5: . 2
         6: 2 .
 
i8 : betti basis( ((FrobPower(J,3)+ideal(f)):ideal (x,y,z))/((FrobPower(J,3)+ideal(f))))
 
o8 = total: 6 2
         3: 1 .
         4: . .
         5: . .
         6: . .
         7: . .
         8: 3 .
         9: . .
        10: . .
        11: . 2
        12: 2 .
 
i9 : betti basis( ((FrobPower(J,4)+ideal(f)):ideal (x,y,z))/((FrobPower(J,4)+ideal(f))))
 
o9 = total: 6 2
         3: 1 .
         4: . .
         5: . .
         6: . .
         7: . .
         8: . .
         9: . .
        10: . .
        11: . .
        12: . .
        13: . .
        14: . .
        15: . .
        16: 3 .
        17: . .
        18: . .
        19: . .
        20: . .
        21: . .
        22: . .
        23: . 2
        24: 2 .
 
i10 :