Complexity of tropical Schur polynomials

( We study the complexity of computation of a tropical Schur polynomial Ts-lambda where lambda is a partition, and of a tropical polynomial Tm-lambda obtained by the tropicalization of the monomial symmetric function m(lambda). Then TS lambda and Tm-lambda 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 Ts-lambda, and an exponential lower bound for Trn(lambda). Also the complexity of tropical skew Schur polynomials is discussed. (C) 2015 Elsevier Ltd. All rights reserved.