Kernels in tropical geometry and a Jordan–Hölder theorem

Abstract

Abstract. A correspondence exists between affine tropical varieties and algebraic objects, following the classical Zariski correspondence between irreducible affine varieties and the prime spectrum of the coordinate algebra in affine algebraic geometry. In this context the natural analog of the polynomial ring over a field is the polynomial semiring over a semifield, but one obtains homomorphic images of coordinate algebras via congruences rather than ideals, which complicates the algebraic theory considerably.In this paper, we pass to the semifield F (λ 1 , . . . , λn) of fractions of the polynomial semiring, for which there already exists a well developed theory of kernels, also known as lattice ordered subgroups; this approach enables us to switch the structural roles of addition and multiplication and makes available much of the extensive theory of chains of homomorphisms of groups. The parallel of the zero set now is the 1-set.(Idempotent semifields correspond to lattice ordered groups, and the kernels to normal convex l-subgroups.)These notions are refined in the language of supertropical algebra to ν-kernels and 1 ν -sets, lending more precision to tropical varieties when viewed as sets of common roots of polynomials. The ν-kernels corresponding to (supertropical) hypersurfaces are the 1 ν -sets of corner internal rational functions. The ν-kernels corresponding to "usual" tropical geometry are the regular, corner-internal ν-kernels.In analogy to Hilbert's celebrated Nullstellensatz which provides a correspondence between radical ideals and zero sets, we develop a correspondence between 1 ν -sets and a well-studied class of ν-kernels of the rational semifield called polars, originating from the theory of lattice-ordered groups. This correspondence becomes simpler and more applicable when restricted to a special kind of ν-kernel, called principal, intersected with the ν-kernel generated by F . We utilize this theory to study tropical roots of finite sets of tropical polynomials.For our main application, we develop algebraic notions such as composition series and convexity degree, along with notions having a geometric interpretation, like reducibility and hyperdimension, leading to a tropical version of the Jordan-Hölder theorem for the relevant class of ν-kernels. IntroductionThis paper is a combination of [29] and [30]. The underlying motive of tropical algebra is that the valuation group of the order valuation (and related valuations) on the Puiseux series field is the ordered group (R, +) or (Q, +) (depending on which set one uses for powers in the Puiseux series), which can also be viewed as the max-plus algebra on (R, +) or (Q, +). This leads one to a procedure of tropicalization, based on valuations of Puiseux series, which takes us from polynomials over Puiseux series to "tropical" polynomials over the max-plus algebra. One of the main goals of tropical geometry is to study the ensuing varieties. Traditionally, following Zariski, in the affine case, one would pass to the ideal of the polynomial alge...

On the dimension of polynomial semirings

“…It is then verified, amongst several other results, that the dimension of a rational function field over an archimedean semifield equals the number of variables. This result is somewhat analogous to the main theorem of the current paper, however neither of the two results imply special cases of the other, since our notion of dimension differs from that of [PR15]. We also note that the results of [PR15] are set in the more general context of "supertropical algebra", but this setting contains the usual semirings as a degenerate special case.…”

“…This result is somewhat analogous to the main theorem of the current paper, however neither of the two results imply special cases of the other, since our notion of dimension differs from that of [PR15]. We also note that the results of [PR15] are set in the more general context of "supertropical algebra", but this setting contains the usual semirings as a degenerate special case. To avoid possible confusion we point out that our terminology differs from that of [PR15], where the authors call every cancellative semiring a domain.…”

“…Remark 2.10. In [PR15] the authors consider a sublattice of all congruences of a rational function semifield, generated by the so-called hyperplane kernels. They define the dimension as the maximum of the length of chains of irreducibles in this particular sublattice.…”

“…We also note that the results of [PR15] are set in the more general context of "supertropical algebra", but this setting contains the usual semirings as a degenerate special case. To avoid possible confusion we point out that our terminology differs from that of [PR15], where the authors call every cancellative semiring a domain. Also in [PR15] kernels refer to the equivalence class of 1 in a congruence (of a semifield) and not to the equivalence class of 0 as in the current paper.…”

“…A different approach to establish a tropical notion of dimension using chains of congruences was taken by L. Rowen and T. Perri in [PR15], which we will briefly recall in Remark 2.10. A common theme in [PR15] and the present work is that to obtain a good notion of dimension one has to bypass the difficulties that come from polynomial semirings having too many congruences. In fact, as we will see in Proposition 2.12, any additively idempotent polynomial semiring in at least 2 variables has infinite chains of congruences with cancellative quotients.…”