# Corrado Zanella
# A condition for scattered linearized polynomials involving Dickson matrices
# Script referred-to in remark 3.7

Elenco:=[]; # The list of prime powers less than 223
for q in [3..222] do
  if IsPrimePowerInt(q) then Add(Elenco,q); fi;
od;

n:=5;
for q in Elenco do #main loop
  qqq:=q^2-q;
  Print("\nComputing q=",q,"...\n");
  O:=0*Z(q);
  U:=Z(q)^0;

  for esp in [1..q-1] do
    b:=Z(q^n)^esp; # N_(q^n/q)(b) will assume all possible nonzero values
    scat:=true; # scat will be true if x^q+bx^q^2 is a scattered polynomial
    for x in GF(q^n) do #seeking for a solution of (6)
      if x<>O then
        w:=AdditiveInverse(U+Inverse(b*x^qqq));
        addendo:=U; # the generic element of the sum in (6)
        somma:=O; # the sum in (6)
        for r in [0..n-1] do
          somma:=somma+addendo; # that is, sum_{i=0}^r w^((q^i-1)/(q-1))
          addendo:=w*(addendo^q); # that is, w^((q^(r+1)-1)/(q-1))
        od;
        if somma=O then scat:=false; Print("non scattered, "); break; fi; # found a solution of (6) - exit from 'x' loop
      fi; # if x<>0
    od; # end of 'x' loop
  if scat then Print("\nFor b=",b," it is scattered???\n\n"); return; fi; # no solution for (6) found
  od; # end of 'esp' loop
od; # end of 'q' loop