Dear RT:

You wrote:
>
> The original note was, “Busy now, will write later.” This seems like two
> sentence fragments, joined by a comma. That seems quite incorrect,
> grammatically speaking.

This is what we might call “notice writing.” This is the kind of thing you see on notices and signs, along the lines of “Wet paint” or “Back at 1:30.” It would either be two sentence fragments, as you wrote, or a single sentence fragment with a compound verb. In everyday speech the missing words would be understood: Either “[I am] busy now, [but I] will write later” or “[I am] busy now [and] will write later.” At best this is conversational; it certainly is not standard.

You replied:
>
> Thank you, but my problem is with the lack of the conjunction before the
> verb ‘will.’ I understand that’s it fine for handwritten notes (most
> everything is), but technically speaking, I didn’t know that one can
> simply
> omit the conjunction and still have the sentence be grammatically correct.
> Is there a rule to support your answer?
>
> Would “I’m somewhat busy, talk later.” be correct as well?
>
> Thanks again!
>
No, to be grammatically correct you are right. You would need to have a conjunction at the least: “I am busy and will talk later.” “I am busy; talk later” would work if the “talk” were imperative. I understood your question to be about a quickly jotted memo or something similar. We would understand it, but, no, it is not standard English.

Sometimes even in formal English or poetry the conjunction may be dropped to get someone’s attention. That figure of speech is called asyndeton.