A classification of the commutative Banach perfect semi-fields of characteristic 1: applications

Abstract

A la mémoire de Tilby. 2 ERIC LEICHTNAM 7.2. Valuation and localization for semi-rings. 38 7.3. Locally semi-ringed spaces. Schemes. 41 References 43 1. Introduction.A semi-ring (R, ⊕, +) of characteristic 1 is a set R endowed with two commutative and associative laws satisfying the following conditions. The law ⊕ is idempotent (e.g. X ⊕ X = X, ∀X ∈ R) whereas the additive law + has a neutral element 0 and is distributive with respect to ⊕. The semi-ring R is said to be cancellative if X + Y = X + Z implies Y = Z. If for any n ∈ N * , X → nX is surjective from R onto itself, then (R, ⊕, +) is called a perfect semi-ring. If (R, +) is a group then (R, ⊕, +) is called a semi-field and we denote it by F rather than R.Morally, the first law ⊕ is the new addition and the second law + is the new multiplication. But, being idempotent, ⊕ is very far from being cancellative.The theory of semi-rings is well developed (see for instance [Go], [Pa-Rh]). It has important applications in various areas: tropical geometry ([Ak-Ba-Ga], [Br-It-Mi-Sh]), computer science, tropical algebra ([Iz-Ro], [Le]), mathematical physics ([Li]). In the references of these papers the reader will find many other interesting works in these areas.Thanks to Connes-Consani, the semi-ring theory of characteristic 1 plays also a very interesting role in Number Theory: [Co-C0], [Co-C1], [Co-C2], [Co-C3]. For instance they constructed an Arithmetic Site (for Q) which encodes the Riemann Zeta Function. They analyzed the contribution of the archimedean place of Q through the semi-field (R, max (, ), +), the multiplicative action of (R + * , ×) on R being an analogue of the Frobenius.In constructing Arithmetic sites of Connes-Consani type for other numbers fields K, Sagnier( [Sa]) was naturally lead to more general semi-fields of characteristic 1. They are associated to some tropical geometry. For example, in the case K = Q[i], [Sa] introduced the set R c of compact convex (polygons) of C which are invariant under multiplication by i. Then he analyzed the contribution of the archimedean place of Q[i] through the semi-field of fractions of the cancellative perfect semi-ring:Furthermore, Connes-Consani ([Co-C0], [Co]) have developed a very interesting theory of Witt vectors in characteristic 1. Following [Co] one defines the analogue of a p−adic metric on a perfect semi-field F in the following rough way (see Section 2 for details). First recall the well-known partial order on F defined by:Next, it turns out that F carries a natural structure of Q−vector space. Assume then the existence of E ∈ F such that for any X ∈ F there exists t ∈ Q + such that −tE ≤ X ≤ tE (see Assumption 1). Denote by r(X) ∈ R + the infimum of all such t ∈ Q + and assume moreover that r(X) = 0 implies X = 0 (see Assumption 2). Then (X, Y ) → r(X − Y ) defines a distance on F ; we say that (F , ⊕, +) is a Banach semi-field if F is complete for this distance.The norm r(X) satisfies also the following inequality of non-archidemean type:∀X, Y ∈ F , r(X ⊕ Y ) ≤ max(r(X), r(Y )) .

Idempotence of finitely generated commutative semifields

“…Let us conclude this introduction by pointing out that the ideas presented in this paper can probably be generalized to the situation of additively divisible semirings. Another very interesting generalization of our results and methods is to apply them to the Banach semifield setting of Leichtnam [26]. This should (hopefully) allow us to generalize and extend his results (e.g., to remove the Assumption 2) -we also plan to study this in the (near) future.…”

“…In number theory, Connes and Consani [7,8] were motivated by the goal of working over the "field of one element" [33] (related to semirings) and extended this viewpoint further with a certain hope of proving Riemann hypothesis. Their work was recently generalized by Leichtnam [26] to cover more general additively idempotent semifields. An interesting direction is also the study of cryptography based on semirings, as developed by Maze, Monico, Rosenthal, Zumbrägel, and others [28,29,38].…”

Commutative $ν$-algebra and supertropical algebraic geometry

“…Tropical semirings pR, max, `q, where R " B, N, Z, are linked to number theory [6,7,8] and arithmetic geometry via the Banach semifield theory of characteristic one [44]. Features of traditional tropical geometry are received as the Euclidean closures of "tropicalization" of subvarieties of a torus pK ˆqn , where K is a non-archimedean algebraically closed valued field, complete with respect to the valuation [14,20,54].…”

Semifields and a theorem of Abhyankar

“…Semirings are a very natural generalization of rings that has been widely studied, not only from purely algebraic perspective, but also for their applications in cryptography, dequantization, tropical mathematics, non-commutative geometry, and the connection to logic via MV-algebras and latticeordered groups [2], [3], [4], [5], [6], [15], [16], [17], [18], [19], [20], and [21]. We refer the reader to the aforementioned works for further history and references.…”