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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 thereal Grassmannian, is the vector obtained by taking the $k$th power of eachentry of $v$ again a vector of Pl\"ucker coordinates? For $k\neq 1$, this istrue if and only if the corresponding matroid is regular. Similar results holdover other fields. We also describe the subvariety of the Grassmannian thatconsists 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 anon-negative integer. We prove that if $\mathcal{A}$ is representable and theunderlying matroid is non-regular, then $\mathcal{A}^k := (E, rk, m^k)$ is notrepresentable. This provides a large class of examples of arithmetic matroidsthat are not representable. On the other hand, if the underlying matroid isregular and an additional condition is satisfied, then $\mathcal{A}^k$ isrepresentable. Bajo-Burdick-Chmutov have recently discovered that arithmeticmatroids of type $\mathcal{A}^2$ arise naturally in the study of colourings andflows on CW complexes. In the last section, we prove a family of necessaryconditions for representability of arithmetic matroids.

On powers of Pl\"ucker coordinates and representability of arithmetic matroids