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I'm a student currently pursuing my masters in computer science. My present area of research is computational algrebraic geometry (theory of grobner bases and tropical geometry).
Afterwards, I'm interested in pursuing my PhD in pure mathematics.
I would like know if I'll be elligible for applying to pure math PhD programs? And if yes, do I have any realistic chance of getting into one?
What else can I do to improve my chances of being accepted?
Short answer: Yes. Absolutely. You are already doing mathematics.
A few bits of advice:
Ask your advisor and other references to specifically address your mathematical depth and maturity in their recommendation letters.
Include a technical summary of your past research, including pointers to ArXiv preprints if possible, in your statement of purpose.
Apply to math departments that employ computational algebraic geometers. Three that come to mind immediately are Saugatu Basu at Purdue, Frank Sottile at Texas A&M, and Bernd Sturmfels at Berkeley.
(I know at least half a dozen former CS grad students who successfully switched into mathematics.)
I've served on graduate admissions committees for a math department in a research university. It really won't matter what your degrees are in, as long as the substance is there, but you need to be careful of several points:
You need recommendations from people the committee is familiar with, preferably at least one or two mathematicians. The more they committee knows about your letter writers, the better. (For example, if someone says you are one of the best students in recent years, is that because you are great or because they give inflated praise to everyone? There's no way of knowing if the committee hasn't seen previous recommendations from this person.) It's also important for your recommenders to have enough of a feeling for math grad school that they can confidently address whether you will be successful; for example, theoretical computer scientists can probably do this better than people in more applied areas of CS. Finally, many people believe recommenders are a little less selective about recommending people for things outside of their own area, so recommendations from other fields are often given less weight.
You need to demonstrate that you have mastered the undergraduate material that is less relevant for computer science. For example, mathematical analysis along the lines of Rudin's Principles of Mathematical Analysis. Even if you know a lot about algebra and discrete math, you might still not be in a position to pass some first-year graduate courses in pure math, or you might discover that you don't enjoy them.
You need a compelling story for why you are changing departments. If you focus on not liking what you are currently doing, or on wanting a fresh start, then it will not go over well. There are much more positive stories, about gradually coming to appreciate that pure math is your real interest, but you should write carefully. This is tantamount to admitting that the last time you entered a graduate program, you didn't really know what you were interested in, and you don't want to leave the committee with any worries that this might still be true. If you can say this honestly, then it might help to say something along these lines: when you finished your undergraduate degree, you were unsure about pure math vs. CS, so you decided to write an extremely mathematical CS master's thesis, and this experience has helped you decide where your real interests are. (Being unsure in the past comes across better than having thought you were sure and then realizing you were wrong.)
(I speak from limited experience, and none on the "reading applications and deciding" side. Technically my undergraduate degree was in a program somewhere between math and CS, but I'd taken a fairly math-oriented slant on it.)
Having a masters degree in a related field certainly doesn't make you ineligible to pursue a PhD in math. The only disadvantage I can see for taking time to do a masters degree in a slightly different area, as opposed to, say, working in industry, is that you may need to work harder to explain why you're changing areas.
(Also you'll have some record of doing research, and you might be judged in part on the quality of that research where an undergrad might be given the benefit of the doubt.)
If you want to continue working in a closely related area that's a bit more in the mathematical direction, it shouldn't be very hard to explain why you've decided to continue that work in mathematics rather than CS. It's a bigger issue if you want to do something very different in math---the worry would be that whatever caused you to lose interest in computational algebraic geometry will happen again. That's not insurmountable, but it will be more of a hurdle to convince the faculty you're really interested enough in the new area to stick out a PhD.
I wouldn't rate your chances as very high, but the following factors might help nevertheless:
Your credentials (where you did your BS/MS from), and what balance of mathematics courses did you take - were there any electives, and of course, how you fared in them.
Since you didn't mention it, I'm assuming you haven't published in any decent mathematical venues - it helps your chances a lot if you had publishable results prior to your application. If you find it difficult to get published, post (interesting and useful) results to arxiv - that would boost your case in any case.
Mathematics is a vast field - it would help a lot if you were to continue along the directions to which you've already had some research background in (and maybe published on), and could get an interested faculty to have a look at your profile.
