You can do a semi-self-arrest using a trekking pole, but only in gentle conditions. As soon as it gets steeper and icier, you will need a real ice axe. The real ice axe has a real pick, and that will dig in much deeper than a bottom tip of a trekking pole.
So, if you are sure that your conditions will be gentle, then you can probably get away with it.
Personally, I would rather have a little more ice axe than what I need rather than too little.