The Macaulay session which supports Example 4 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 J=(x^2,x*z,y^2,y*z,x*y+z^2) and f=x^3+y^3+z^3.


We learn:

e  socle degrees
0  2:1
1  4:7
2  9:12
3  19:12
4  39:12

kustin@berry 6 % 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^2,x*z,y^2,y*z,x*y+z^2);
 
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))/((J+ideal(f))))
 
o5 = total: 6 1
         1: . 1
         2: 6 .
 
i6 : betti basis( ((FrobPower(J,1)+ideal(f)):ideal (x,y,z))/((FrobPower(J,1)+ideal(f))))
 
o6 = total: 13 7
         3:  1 7
         4: 12 .
 
i7 : betti basis( ((FrobPower(J,2)+ideal(f)):ideal (x,y,z))/((FrobPower(J,2)+ideal(f))))
 
o7 = total: 18 12
         3:  1  .
         4:  .  .
         5:  .  .
         6:  .  .
         7:  .  .
         8:  5 12
         9: 12  .
 
i8 : betti basis( ((FrobPower(J,3)+ideal(f)):ideal (x,y,z))/((FrobPower(J,3)+ideal(f))))
 
o8 = total: 18 12
         3:  1  .
         4:  .  .
         5:  .  .
         6:  .  .
         7:  .  .
         8:  .  .
         9:  .  .
        10:  .  .
        11:  .  .
        12:  .  .
        13:  .  .
        14:  .  .
        15:  .  .
        16:  5  .
        17:  .  .
        18:  . 12
        19: 12  .
 
i9 : betti basis( ((FrobPower(J,4)+ideal(f)):ideal (x,y,z))/((FrobPower(J,4)+ideal(f))))
 
o9 = total: 18 12
         3:  1  .
         4:  .  .
         5:  .  .
         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:  5  .
        33:  .  .
        34:  .  .
        35:  .  .
        36:  .  .
        37:  .  .
        38:  . 12
        39: 12  .
 
i10 :