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?
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.