Subset $k$-sum problem has been well studied as a fixed parameter version of subset sum.
What is known about the analogous Subset $k$-product problem which is the fixed parameter version of subset product?
I am interested in the case where the base field has characteristic $0$ and not $0$ and large of order $O(2^n)$ where $n$ is the number of elements as input to the problem.
Is there a faster than $n^{O(k)}$ algorithm for this problem?