Emil's Centurion boost
Influence COULD be the right word to use, but to answer your question, here's what happens.
When you get a Centurion's core, you also obtain a skill of the same name. That skill, when equipped, takes a certain arte and gives it the elemental attribute of the Centurion.
Other than getting type advantages/disadvantages, the only other useful part I've found of this, is being able to fill up that meter in the bottom left with on element so you can capture a monster.