If $A = (\alpha\_{ij}) \in \mathbb{C}^{nxm}$ we have simple algorithms by which to determine $\mathrm{rank}(A)$. However, is there a polynomial $f \in \mathbb{C}[\alpha\_{ij}]$ where $f \colon \mathbb{C}^{nxm} \to \mathbb{N}$ such that $f(A) = \mathrm{rank}(A)$?


In general, is it possible to determine if a particular algorithm may be expressed as a polynomial or other more general function?



Rank(A) and other algorithms as a polynomial
The preimage of a natural number under a polynomial map is always closed, yet the set of matrices with rank one is not closed. For example, the sequence $(A\_n)\_{n\geq1}$ in $M\\_2(\mathbb C)$ with $A\_n=\left(\begin{smallmatrix}1/n&0\\\0&0\end{smallmatrix}\right)$ has all its items of rank one, yet its limit has rank zero.


