I made observation than AI always expand his influence zone more quickly than me (maybe because I play at Impossible).

Anyway, my question is :

- What determine the grow of my influence zone ?

- Is there specific actions, techs to search, or any trick making I could focus on growing more this influence zone ?

The grow of influence zone is determined by the influence production of its system. For my part when i want to have a good influence zone i choose one system (generally with a lot of productivity and pop slot) and focus influence boost on it. The culture center wonder in interspecies HR is very good, and counsellors are very strong for boost influence (especially horatio counsellors). Stack everything in one system as soon as possible and get ready to see your influence turn insane. I once got a system with 1700 influence/turn. My influence territory was the entire galaxy pretty fast x).

I learned a lot here, 

big tank you !!!!