PhD in pure mathematics for a student in computer science
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&amp;M, and Bernd Sturmfels at Berkeley. 


(I know at least half a dozen former CS grad students who successfully switched into mathematics.)