Is it impolite to ask to which journal a preprint was submitted? Is it considered impolite to ask the author of a preprint about the journal to which the paper was submitted? The specific reason why I would be asking is that this would help me gauge how high to aim with my own related preprint.

To give a few more details: I recently run into a preprint by someone, let's call him Smith, who obtained a partial result on a problem that struck me as interesting. After giving it a little thought, I stumbled into some ideas that solve the problem completely* and wrote to Smith to ask him about his thoughts. As it turned out, Smith has independently solved the general case as well, by entirely different methods, and we ended up posting our preprints online days apart. Now, I'm trying to figure out how interesting the new result actually is. As is sometimes the case in (certain combinatorial branches of) mathematics, it's somewhat hard to tell, and a significant part of motivation is that people have been thinking about the problem and didn't come up with a solution. Ideally, if Smith's first preprint had already been published, I would send my work to a journal that's one notch higher, or if I knew where the first preprint is sent I would send my work to a journal of the same caliber. Also, I would probably feel a lot of regret if Smith published his paper in a journal more prestigious than me simply because I didn't aim high enough (if the referees find his argument more interesting than mine that's fine of course, I'd just feel regret about not giving it a shot). On the other hand, I'm worried that asking about where he sent his preprints would be considered impolite - it could be considered too intrusive, and there are good reasons why people don't publicise which journals they have submitted to (some discussed here and here on this SE). Then again, maybe it would not - after all, people are generally willing to share more in private communication than publicly. If it matters, I have met Smith on several occasions before and we talked a bit but not a lot; we're on similar stages of our careers with him being slightly senior.



*) Well, almost completely, depending on how you define the problem.
Yes, I think this question is impolite.  The submission history of an article is usually considered a private matter, because if the paper doesn't appear in the journal where it was submitted, one can infer that it was rejected.  It's a little like asking about their sex life.  If someone I knew well asked me where I had submitted a paper, I might tell them - but if a stranger or casual acquaintance asked me, my reaction would be "That's none of your damn business."   (My actual reply might be a more polite version like "I'd rather not discuss that.")

I suppose that, as Anonymous Physicist says, you could ask Smith if he has any advice on where to send such a paper.  But since you seem to be competitors, he might not be inclined to help you out in this regard - in fact, if he does offer a suggestion, you may not really want to trust it.

It would also be reasonable to contact Smith and propose a joint paper, though of course he is free to decline.