Complexity of tropical Schur polynomials

We study the complexity of computation of a tropical Schur polynomial Tsλ where λ is a partition, and of a tropical polynomial Tmλ obtained by the tropicalization of the monomial symmetric function mλ. Then Tsλ and Tmλ coincide as tropical functions (so, as convex piece-wise linear functions), while differ as tropical polynomials. We prove the following bounds on the complexity of computing over the tropical semi-ring (R,max,+): • a polynomial upper bound for Tsλ and • an exponential lower bound for Tmλ. Also the complexity of tropical skew Schur polynomials is discussed.