The field is mathematics (functional analysis). As we get ready to post a paper on arxive, we realized that one of the results here disproves a conjecture proposed in another paper.
Assuming the author of that conjecture is still around, is it good practice to communicate with the author first before we post our paper?
Conversely, would it be impolite/rude to post our paper without notifying the conjecture's author?
Personally, I'd be more than happy to be notified and to see an early copy of your work. Among other things it might save me some work and would put my mind to rest in any case.
Assuming you cite the other author then it would be polite to notify them so that it isn't a surprise when your work appears.
You did cite them, I hope.
Note that sometimes such things happen when the new author doesn't know about the earlier work so it isn't always possible to notify the original author. But since you noticed, yes, let them know.
If you were claiming that someone had made a mistake in their paper (in mathematics), then in that situation you should notify them beforehand and give them a chance to rebut your claim. (Keep in mind one of the possibilities would be that the claims in your paper are correct but you didn't understand the claims in their paper.) However, a refuted conjecture isn't a mistake, because a conjecture isn't a claim at all, only an informed guess (or, in some cases, a provocative way to ask a question). Hence, you aren't under some obligation to inform them before making the paper public. You certainly can if you want, and they could have some further information, for example further context for the conjecture, that would help you produce a better paper.
However, if you're asking about notifying them roughly simultaneously with putting it on the ArXiv, I don't see why you're asking. Mathematicians have a hard enough time getting anyone to read their papers as it is; by some measures of what it means to 'read', the median paper is probably read by fewer than one person not including authors and referees. The person who proposed this conjecture is naturally someone who is likely to be interested in your paper, and of course it is to your advantage if they know about it. If this is a well-connected senior mathematician who organizes conferences and workshops and might help you publicize your work further, even better. Without being obnoxious about it (and letting someone know that you have resolved their conjecture is definitely not obnoxious), you want to publicize your work to interested parties as much as possible.
Also, this person is a likely referee for your paper, and it will speed the process along if they happen to have already read your preprint prior to being asked to referee!
Yes, you should notify the author of the conjecture that you believe yourself to have rebutted.
In addition to the other reasons given, that other author, being human, will have some emotional investment in the conjecture, and thus will be motivated to give your rebuttal the highest level of scrutiny.
After all, it may be that you are the one in error and the other author's reasoning is in fact quite sound, and if this is indeed the case it is best for everybody involved to expose the error as soon as possible, wherever that error may be.
