A new definition of prime congruences in additively idempotent semirings is given using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences it is shown that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. A complete description of prime congruences is given in the polynomial and Laurent polynomial semirings over the tropical semifield T, the semifield Z max and the two element semifield B. The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. It is then shown that the radical of every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension 1. An improvement of a result from [BE13] is proven which can be regarded as a Nullstellensatz for tropical polynomials. 2010 MSC: 14T05 (Primary); 16Y60 (Primary); 12K10 (Secondary); 06F05 (Secondary) Keywords: idempotent semirings, tropical polynomials 2 Prime congruences of semirings In this paper by a semiring we mean a commutative semiring with multiplicative unit, that is a nonempty set R with two binary operations (+, ·) satisfying:(i) (R, +) is a commutative monoid with identity element 0

We compute toric degenerations arising from the tropicalization of the full flag varieties Fℓ 4 and Fℓ 5 embedded in a product of Grassmannians. For Fℓ 4 and Fℓ 5 we compare toric degenerations arising from string polytopes and the FFLV polytope with those obtained from the tropicalization of the flag varieties. We also present a general procedure to find toric degenerations in the cases where the initial ideal arising from a cone of the tropicalization of a variety is not prime.

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From any monoid scheme X (also known as an F 1 -scheme) one can pass to a semiring scheme (a generalization of a tropical scheme) X S by scalar extension to an idempotent semifield S. We prove that for a given irreducible monoid scheme X (satisfying some mild conditions) and an idempotent semifield S, the Picard group Pic(X) of X is stable under scalar extension to S (and in fact to any field K). In other words, we show that the groups Pic(X) and Pic(X S ) (and Pic(X K )) are isomorphic. In particular, if X C is a toric variety, then Pic(X) is the same as the Picard group of the associated tropical scheme. The Picard groups can be computed by considering the correct sheaf cohomology groups. We also define the group CaCl(X S ) of Cartier divisors modulo principal Cartier divisors for a cancellative semiring scheme X S and prove that CaCl(X S ) is isomorphic to Pic(X S ).