Fast differential eleminination in C: The CDiffElim environment. We introduce the CDiffElim environment, written in C, and an algorithm developed in this environment for simplifying systems of overdetermined partial differential equations by using differentiation and elimination. This environment has strategies for addressing difficulties encountered in differential elimination algorithms, such as exhaustion of computer memory due to intermediate expression swell, and failure to complete due to the massive number of calculations involved. These strategies include low-level memory management strategies and data representations that are tailored for efficient differential elimination algorithms. These strategies, which are coded in a low-level C implementation, seem much more difficult to implement in high-level general purpose computer algebra systems. ...