>>51151138There are more than eight of those battlefields, though.
There are 7 at 7 of the gym buildings. There is one on the beach near where you start the game. There is another in the brown area on the right side of the map. There appear to be two in Mesagoza (the blue one to the left and the central square appears to be one as well). There's another on top of that colosseum I pointed out.
If each battlefield was a gym, there would be 11 gyms.
Like I said, I think the gyms are those towers shown in the trailer and on the website when they talk about the gym challenge. I think the gym leaders will take you outside to the nearby battlefields to battle you outside.