Symmetric polynomials in tropical algebra semirings

Tropical Semirings

It is a well-known fact that the boolean calculus is one of the mathematical foundations of electronic computers. This explains the important role of the boolean semiring in computer science. The aim of this paper is to present other semirings that occur in theoretical computer science. These semirings were baptized tropical semirings by Dominique Perrin in honour of the pioneering work of our ...

Nonfinitely Based Tropical Semirings

Equational theories of tropical semirings

This paper studies the equational theory of various exotic semirings presented in the literature. Exotic semirings are semirings whose underlying carrier set is some subset of the set of real numbers equipped with binary operations of minimum or maximum as sum, and addition as product. Two prime examples of such structures are the (max,+) semiring and the tropical semiring. It is shown that non...

Linear independence over tropical semirings and beyond

We investigate different notions of linear independence and of matrix rank that are relevant for max-plus or tropical semirings. The factor rank and tropical rank have already received attention, we compare them with the ranks defined in terms of signed tropical determinants or arising from a notion of linear independence introduced by Gondran and Minoux. To do this, we revisit the symmetrizati...

Symmetric Polynomials

f(T1, . . . , Tn) = f(Tσ(1), . . . , Tσ(n)) for all σ ∈ Sn. Example 1. The sum T1 + · · ·+ Tn and product T1 · · ·Tn are symmetric, as are the power sums T r 1 + · · ·+ T r n for any r ≥ 1. As a measure of how symmetric a polynomial is, we introduce an action of Sn on F [T1, . . . , Tn]: (σf)(T1, . . . , Tn) = f(Tσ−1(1), . . . , Tσ−1(n)). We need σ−1 rather than σ on the right side so this is a...