I wonder if you can solve the following problem in general: Let $f$ be a polynomial in $K[x]$, where $K$ is a field. Find a polynomial $g$ such that $fg$ has minimal number of terms (not zero).
If you restrict the maximum possible degree of $g$, you can solve the problem easily (although with a lot of computational effort, if I see it correctly).
What can be said for arbitrary multiples? Can one somehow restrict the degree of a shortest multiple by the degree and the size of the coefficients of $f$?
If anyone knows more about this, I would be very happy.
(The question has a connection to the following thread: Minimum number of terms resulting from the product of two polynomials with a given number of terms)
