Polynomials on Affine Manifolds

For a closed affine manifold $M$ of dimension $m$ the developing map defines an open subset $D(\tilde M) \subset {{\mathbf {R}}^m}$. We show that $D(\tilde M)$ cannot lie between parallel hyperplanes. When $m \le 3$ we show that any nonconstant polynomial $p:{{\mathbf {R}}^m} \to {\mathbf {R}}$ is unbounded on $D(\tilde M)$. If $D(\tilde M)$ lies in a half-space we show $M$ has zero Euler characteristic. Under various special conditions on $M$ we show that $M$ has no nonconstant functions given by polynomials in affine coordinates.