Symmetric polynomials in tropical algebra semirings

The growth of tropical geometry has generated significant interest in the tropical semiring in the past decade. However, there are other semirings in tropical algebra that provide more information, such as the symmetrized ( max ⁡ , + ) , Izhakian's extended and Izhakian–Rowen's supertropical semirings. In this paper we identify in which of these upper-bound semirings we can express symmetric polynomials in terms of elementary ones. We show that in the case of idempotent semirings we can do this precisely when the Frobenius property is satisfied, that in the case of supertropical semirings this is always possible, and that in non-trivial symmetrized semirings this is never possible. Our results allow us to determine the tropical algebra semirings where an analogue of the Fundamental Theorem of Symmetric Polynomials holds and to what extent.