  We find a semi-algebraic description of the Minkowski sum $\mathcal{A}_{3,n}$
of $n$ copies of the bounded twisted cubic $\{(t,t^2,t^3)\mid -1\leq t\leq 1\}$
for each integer $n\geq3$. These descriptions provide efficient membership
tests for the sets $\mathcal{A}_{3,n}$. These membership tests in turn can be
used to resolve some instances of the underdetermined matrix moment problem,
which was formulated by Michael Rubinstein and Peter Sarnak in order to study
problems related to $L$-functions and their zeros.
