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# Computes the size of the kernel of the map I^n/I^(n+m) -> I^n*p^l/I^n*p^l+m; x -> x^(p^l), cf. Hertweck-Soriano Section 2.7
BindGlobal("KernelSizePowerMap", function(T, n, m, l)
local Res, p, max, v0, pos, tup, s, v, w, i, posm, t, count, k;
Res := 0;
k := T.fld;
p := Characteristic(k);
max := Maximum(T.wgs);
if max < n*(p^l)+m-1 then
Print("Weight of table is too small, recompute for weight ", n*(p^l)+m, "\n");
return fail;
fi;
pos := Filtered([1..Size(T.wgs)], x -> n <= T.wgs[x] and T.wgs[x] <= n+m-1);
s := Size(pos);
posm := Filtered([1..Size(T.wgs)], x -> T.wgs[x] < n*(p^l)+m);
t := Size(posm);
v0 := ListWithIdenticalEntries(t, 0*Z(p));
count := 0;
#Act on "projective" space to save time, hence the s-1
for tup in IteratorOfTuples(k, s-1) do
count := count + 1;
if count mod 1000 = 0 and InfoLevel(InfoModIsom) >= 1 then
Print("Step ", count, " of ", NrTuples(k,s-1), "\n");
fi;
#Case last entry is 0
v := ListWithIdenticalEntries(T.dim, 0*Z(p));
for i in [1..s-1] do
v[pos[1]-1+i] := tup[i]*(Z(p)^0);
od;
w := v;
for i in [1..p^l-1] do
w := MultByTable(T, w, v);
od;
if w{[1..t]} = v0 then
Res := Res + 1;
fi;
#Case last entry is 1
v := ListWithIdenticalEntries(T.dim, 0*Z(p));
v[pos[1]-1+s] := Z(p)^0;
for i in [1..s-1] do
v[pos[1]-1+i] := tup[i]*Z(p);
od;
w := v;
for i in [1..(p^l)-1] do
w := MultByTable(T, w, v);
od;
if w{[1..t]} = v0 then
#As any multipel of w is also in the kernel
Res := Res + Size(k)-1;
fi;
od;
return Res;
end);
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