Well, of course the young mathematician should simply discuss the
matter with the senior mathematician and perhaps the student until
they can come to an agreeable arrangement. My advice is that they
should all talk about it. Co-authorship is a matter upon which all authors must agree. What other answer could there be?

If it seemed that the professor or the student had little or no
contribution, then the young mathematician should say so and
inquire why the professor should be co-author, or why the student
should be co-author. If there was not sufficient contribution, then
the young mathematician should simply say so and there should be a
discussion about it. Perhaps the senior mathematician will point
out that the contribution was greater than realized, or that there
were other aspects of the collaboration of which the young
mathematician is not aware. Or perhaps not, and the senior
mathematician will agree that the young mathematician should
proceed solo.

But apart from the particular situation described in the post, let
me now mention several further reactions that I have more generally
to the issues about co-authorship that this question raises.

The first and most important thing to say is that collaborative
research is one of the great joys of mathematical life, and I
strongly recommend it. To discuss a mathematical idea with another
mathematician, who can understand what you are saying and who has
thought deeply about the very same topic, gives enormous
satisfaction and meaning to one's life as a mathematician.
Collaborative research is our mathematical social life. For my own part, I am
thankful on today, Thanksgiving Day, for the opportunity that I
have had to interact with all my collaborators; I have
learned so much from them. (See the list of my
collaborators.)

Therefore, my advice is that one should seek out collaborations
wherever they are to be found. Often, after one has proved a
theorem, then in joint work it becomes much better, strengthened or
simplified, or a collaborator finds new applications or uses. If
someone asks a question and you answer it, then perhaps the
mathematics is not yet finished, but only begun. Aren't there
further natural questions arising from the result or its proof?
This could be the beginning of a collaboration rather than the end
of one.

Another part of my view is that one should be relaxed about
collaboration and co-authorship. Except in extraordinary cases, the
stakes are low. A mathematician seems to get basically as much
respect and credit for a result, whether or not there are
co-authors on the paper, and so I question whether there really is
any meaningful "dilution" as mentioned in the post. It is simply no
big deal to have co-authors or not.

Therefore, why not be generous? If someone has made a contribution
to your project, even when the contribution is light, then invite
them as co-author. Few mathematical collaborations are perfectly
balanced contributions, and in most collaborations one person has
had a more important insight or made a larger contribution than the
other. But so what? Perhaps the co-author invitation will be
declined, and that is fine, or perhaps they will join and then
proceed to make your result even better. I have had many
collaborations where at first we had a result, which seemed fine
and complete, but then in writing the joint paper we were able
together to improve the result or give further applications, which
wouldn't have happened without the joint interaction. I think you
will often be surprised.

Another point, as I mentioned in the comments, is that asking a
good question in my view is often sufficient for co-authorship. I
have several joint papers that arose from someone asking me a
question (in some cases on MathOverflow), which I answered, and
then asked them to join as co-author. And I've had some the other
way around as well. I find it more natural in such a case, however,
for the theorem-prover to be suggesting the idea of co-authorship,
rather than the question-asker, which is part of why the situation
in the OP seems wrong to me.

Another point is that it sometimes happens that person A, perhaps a
junior person, asks a question that person B, perhaps a senior
person, answers, settling the question; but the situation is that
person A simply cares more about it or has a stronger vision of
what the result can become than person B, who is not as interested.
In this case, the solution is that person A should do the work of
writing the paper, with person B as co-author, even though the
result may have been due to person B. The point is that person B
would not bother to write this paper on their own, but the joint
authorship brings the mathematical paper into existence. The result
can be a great paper, and I know of many papers following this
pattern.

In summary, pursue collaborations; be generous about co-authorship;
be relaxed about co-authorship; enjoy joint mathematical
interaction; make great mathematics.

the situation that you have described but for which I do not have any sight of the actual evidence. en.wikipedia.org/wiki/Rashomon_effect $\endgroup$ – Yemon Choi Nov 23 '17 at 21:23