On Computations with Integer Division

We consider computation trees (CTs) with operations $S \subset {+,-, *, DIV, DIV_c}$, where $DIV$ denotes integer division and $DIV_c$ integer division by constants. We characterize the families of languages $L \subset N$ that can be recognized over ${+,-, DIV_c}$ and ${+,-, *, DIV}$, resp. and show that they are identical. Furthermore, we prove lower bounds for CT's with operations ${+,-, DIV_c}$ for languages $L \subset N$ which only contain short arithmetic progressions. We cannot apply the classical component counting arguments as for operation $S \subset {+,-, *, ./.}$ because Geometry of Numbers about arithmetic progressions on integer points in high-dimensional convex sets for our lower bounds.