Commutative $ν$-algebra and supertropical algebraic geometry

Abstract

This paper lays out a foundation for a theory of supertropical algebraic geometry, relying on commutative ν-algebra. To this end, the paper introduces q-congruences, carried over ν-semirings, whose distinguished ghost and tangible clusters allow both quotienting and localization. Utilizing these clusters, g-prime, g-radical, and maximal q-congruences are naturally defined, satisfying the classical relations among analogous ideals. Thus, a foundation of systematic theory of commutative ν-algebra is laid. In this framework, the underlying spaces for a theoretic construction of schemes are spectra of gprime congruences, over which the correspondences between q-congruences and varieties emerge directly. Thereby, scheme theory within supertropical algebraic geometry follows the Grothendieck approach, and is applicable to polyhedral geometry.

Subordinate quadratic forms and isometric maps over semirings

“…for all x ∈ V. 8 We claim that ρ is a λ-scalar extension of V. Let ϕ : V → W ′ be any λ-linear map of V. Then ϕ • π : V → W ′ is again λ-linear. Since ρ is a λ-scalar extension of V , there exists a unique R ′ -linear map ψ :…”

Stratifications of the ray space of a tropical quadratic form by Cauchy-Schwartz functions

“…Supertropical semirings carry a rich algebraic structure [2,3,6,15,16] and provide the underlying structure of our framework. A supertropical semiring is a unital semiring R with idempotent element e := e + e = 1 + 1 such that, for all a, b ∈ R, a + b ∈ {a, b} whenever ea = eb and a + b = ea otherwise.…”

Amalgamation and extensions of summand absorbing modules over a semiring

“…Example 11.8. Let R = (R, G, ν) be a supertropical semiring [14], or more generally a ν-semiring [4], where G is a bipotent subsemiring of R and ν is a projection R → G, i. When R is a supertrpical semifield [3,14], i.e., T := R \ G is an abelian group and the restriction ν| T : T → G is onto, the subsemiring G is totally ordered and a + b ∈ {a, b} whenever ν(a) = ν(b).…”