What if you did some exploratory work on the solution at the last company you worked for ... are there potential IP issues that could crop up?

Yes, of course. (And I believe YC already asks a question about that.) Employers regularly claim intellectual property rights to the inventions of their employees. But issues coming up does not equal issues being unsurmountable.

First of all, many jurisdictions make it hard for employers to try to control things you invent on your own time without using their facilities -- which doesn't keep employers from trying.

Secondly, even if your employer arguably does control the IP you need, that doesn't mean they won't give it up. Woz famously realized that HP had a claim to the Apple I and offered to let them have it; HP thought of it as more of a hobby than a business and the rest is history.

Universities and other research institutes routinely let employees spin off companies, usually in exchange for some share (although, anecdotally, it seems that universities that are new to the game often ask for more than universities that already have strong spin-off cultures).

In practice the test is where lines of code got written. If all your startup's code was written after you quit your job, you're usually ok.