Can the verb ある be used with the grammar ~せずに?

Question

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!

Answer 1

ある and ~せずに can not be used together.

Yes.
Grammatically ある + ずに is あらずに, but it's never used.

* To make verb + ずに, conjugate the verb to 未然形 and add ずに.
未然形 of ある is あら, so technically you can make up あらずに, but as I said above, actually never be used.

Ringil mentioned あらず, but I think it's not same as other verbs such as 食べず(に).

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To translate "without" into Japanese, なく and 名詞 + なし work in some cases.

I did my homework without any problem
問題なく宿題を終わらせました。

I couldn't have done it without you.
あなたなしでは成し遂げられませんでした。

The answer below seems to explain why あらずに doesn't work while ない-like words do.

「ある」と「ない」は動詞か形容詞か…

Answer 2

The form corresponding to the negative of ある as an adverb is simply なく, just as the negative of ある as a verb is ない.

For instance, you can say 問題なく解決しました to mean "I resolved it without any issues."