Let $C$ be an arithmetic circuit that represents a polynomial $f\in\mathbb K[x_1,\dotsc,x_n]$, with the promise that $f$ has at most $k$ nonzero terms. What is (known about) the complexity of computing $f$ in its sparse representation, given $C$?
I am interested in deterministic and randomized complexity, and in the link with PIT. In particular, does the promise that $f$ is sparse imply good algorithms? A priori, I am more interested in the case of $\mathbb K$ being some finite field, though results over other fields may be relevant.