Semiring Congruences and Tropical Geometry

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Johns Hopkins University
One of the main motivations and inspirations for this thesis is the still open question of the definition of geometry in characteristic one. This is geometry over a structure, called an idempotent semiring, in which 1+1 = 1. While mathematicians have studied semirings for many years, these structures have only recently ignited interest in algebraic geometry, more precisely tropical geometry. This is geometry over a particular idempotent semiring - the tropical semifield. Furthermore, semirings have important number theoretic applications which appear in the work of A. Connes and C. Consani which is focused on finding a new approach to the Riemann hypothesis. We define the prime spectrum of a commutative semiring. Since ideals do not retain their distinguished role in the theory of semirings, the points of this spectrum correspond to certain congruence relations, which we call prime congruences. Motivated by tropical geometry, the key theme of our work is to study the prime spectrum of tropical polynomial semirings, but many of the results presented here apply to any additively idempotent semiring as well. The class of prime congruences which we introduce turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences, we show that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. We give a complete description of prime congruences in the polynomial and Laurent polynomial semirings over the tropical semifield, the subsemifield of integers of the tropical semifield and the Boolean semifield. 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. We show 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. Using this setup we prove one of the main results of this thesis - we improve on a result of A. Bertram and R. Easton which can be regarded as a Nullstellensatz for tropical polynomials. The remaining results are centered about the concept of Krull dimension. We prove that for any idempotent semiring A, we have that dim A[x] = dim A + 1. In the case when we work over the tropical semifield, we relate the dimension of a tropical variety (which is just a polyhedral complex) to our Krull dimension. This shows the relevance of our notion in the context of the standard framework of tropical geometry.
semiring, semifield, tropical variety, prime congruence, radical congruence, krull dimension, characteristic one geometry