  The first problem we investigate is the following: given $k\in
\mathbb{R}_{\ge 0}$ and a vector $v$ of Pl\"ucker coordinates of a point in the
real Grassmannian, is the vector obtained by taking the $k$th power of each
entry of $v$ again a vector of Pl\"ucker coordinates? For $k\neq 1$, this is
true if and only if the corresponding matroid is regular. Similar results hold
over other fields. We also describe the subvariety of the Grassmannian that
consists of all the points that define a regular matroid.
  The second topic is a related problem for arithmetic matroids. Let
$\mathcal{A} = (E, rk, m)$ be an arithmetic matroid and let $k\neq 1 $ be a
non-negative integer. We prove that if $\mathcal{A}$ is representable and the
underlying matroid is non-regular, then $\mathcal{A}^k := (E, rk, m^k)$ is not
representable. This provides a large class of examples of arithmetic matroids
that are not representable. On the other hand, if the underlying matroid is
regular and an additional condition is satisfied, then $\mathcal{A}^k$ is
representable. Bajo-Burdick-Chmutov have recently discovered that arithmetic
matroids of type $\mathcal{A}^2$ arise naturally in the study of colourings and
flows on CW complexes. In the last section, we prove a family of necessary
conditions for representability of arithmetic matroids.
