Closing the interpolation-algebra generating-rank gap

Close the gap between the linear lower bound and the exponential upper bound on the generating rank of interpolation algebras for $\mathcal T_k$, or construct an interpolation algebra with a substantially smaller generating rank.

Background

For the class $\mathcal T_k$ of tropical polynomials with tropical decomposition width at most $k$, the paper establishes a linear lower bound on the generating rank of any interpolation algebra and an upper bound of $e^{(1+o(1))k}$. The authors note that the cyclic tropical polynomial construction attains the latter bound within that construction, so improving the upper bound polynomially would require a different algebraic construction.