As a new researcher, I am in the following situation in mathematics research:
I read paper X, a short paper published in a low-mid tier journal, and found a way to improve and extend the result. The technique I used to extend the result is a different approach to the problem, but not that mathematically technical. However, even though the mathematical extensions are (arguably) trivial, the extensions open the door to a much broader approach to my field. They also open the door up for interesting simple examples that were previously unable to be produced.
I am currently writing up my findings of the original extension in a paper. In this paper, I correctly reference paper X when necessary. From an ethical point of view, it is crystal clear what I did and didn’t do.
1) Is it bad to compare to one single paper often in a paper?
2) What is a good way to tell whether a result is incremental or not?
3) Some of the arguments in paper X need slight modifications under my extension. Is it okay to repeat some of their ideas in my proofs (with clear citation/credit of course)?
Another problem arising from 2): I have another, much more interesting result because of this extension that I have not published. However, this result moves in a different mathematical direction and therefore, I sort of want to write a separate paper on it. I am faced with the following dilemma. I could reasonably do one of the following:
I) Write a longer paper and work to bring the ideas together.
Pros: My separation from paper X is much more obvious and seen as less incremental.
Cons: I mix two different ideas and risk the paper telling too many stories.
II) Vaguely hint at the second result in the first paper. If I do this, the problem is that if my hint is too vague, it will make my first result seem uninteresting. If it is too specific, I risk showing someone else my idea and having them “beat me to the punch.”
Pros: Sticks to one story, but provides additional motivation.
Cons: May make the first result seem too weak.
I apologize if this post is nonspecific, but I imagine others have faced this problem in their relative fields. How did you resolve this dilemma? I realize I have to figure out the answers myself, but how did you figure out the answers to these questions when you were facing this dilemma yourself?