In the definition of a ployA tail it says "as a string of length 10 or greater containing only 'A' or 'N' if you erase all unwanted characters then some 'A's and 'N's that weren't together before might come together. Also note that you probably want to match against [AN]{10,} so that if there are more than 10 A's or N's in a row the match does not fail. Also, MiamiGenome wanted the filenames. This modified version of your code might work a little better:

# if your file extension is not .txt change it to whatever is approria
+te
while (my $filename=<*.txt>){
  # Open the file and slurp the contents to a string.
  open FILE, $filename || die "Cannot open '$filename' for reading: $!
+\n";
  my $file = do { $\ = undef; <FILE> };
  close FILE;

  # If a 'polyA' sequence is found print the file name. 
  if ($file =~ /[AN]{10,}/)
  {
    print "$filename has a polyA tail sequence\n";
  }
}
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