I have been researching on the internet but I can't find any source where they explain how to use ~せずに (without doing) with the verb ある. I also tried several Japanese conjugators out there, but they do not include the conjugation ず, so it didn't help either.
According to the rules of conjugation for ずに, my guess is it would conjugate:
食べる → 食べない → 食べ
ない → 食べずに
ある → ない →
ない → ずに
But it seems really odd to just leave ずに without any root. For example:
I did my homework without any problem. → 問題がずに宿題をした。
Moreover, since ある does not appear in the explanations of the grammar ~せずに I've found so far, my gut feeling is that ある and ~せずに can not be used together, but I really don't know. Can someone confirm this, please?
Thank you for your help!