Manfred Knebusch

Universität Regensburg | UR · Department of Mathematiks

Classes of an equivalence relation on a module $V$ over a supertropical semiring, called rays, carry the underlying structure of 'supertropical trigonometry' and thereby a version of convex geometry which is compatible with quasilinearity. In this theory, the traditional Cauchy-Schwarz inequality is replaced by the CS-ratio, which gives rise to spe...

The starting point and basic notion for the entire book is the general concept of orderings of arbitrary fields. Conceived by Artin and Schreier in their foundational 1927 paper, it was successfully used by Artin in his solution of Hilbert’s 17th Problem in the same year. Real closed fields are introduced and it is shown that they have the same alg...

In this short chapter, a number of important developments and advances are summarized that mostly occurred after the 1989 publication of Einführung in die reelle Algebra, and that are directly related to topics covered in Chaps. 1, 2 and 3.

A detailed introduction to valuation theory of fields is given in this chapter. Valuations are presented from three different points of view: valuation maps, valuation rings, and places. Valuations are naturally associated with orderings, since every convex subring of an ordered field is a valuation ring. The precise relationship between the orderi...

The focus of this chapter is on the real spectrum of a commutative ring—discovered in 1979 by M.F. Roy and M. Coste—and its elementary properties. Most of the time the aim is to be as general as possible and consider arbitrary commutative rings. However, special features arise in the “geometric setting” where coordinate rings of affine varieties ov...

Classes of an equivalence relation on a module V over a supertropical semiring, called rays, carry the underlaying structure of "supertropical trigonometry" and thereby a version of convex geometry which is compatible with quasilinearity. In this theory the traditional Cauchy-Schwarz inequality is replaced by the CS-ratio which gives rise to specia...

This paper expands the theory of quadratic forms on modules over a semiring [Formula: see text], introduced in [11–13], especially in the setup of tropical and supertropical algebra. Isometric linear maps induce subordination on quadratic forms, and provide a main tool in our current study. These maps allow lifts and pushdowns of quadratic forms on...

The paper expands the theory of quadratic forms on modules over a semiring R, introduced in [12]-[14], especially in the setup of tropical and supertropical algebra. Isometric linear maps induce subordination on quadratic forms, and provide a main tool in our current study. These maps allow lifts and pushdowns of quadratic forms on different module...

Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a ‘supertropical trigonometry’ and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy–Schwarz inequality. CS-functions that emerge from the CS-ratio are a useful tool that h...

A submodule [Formula: see text] of [Formula: see text] is summand absorbing, if [Formula: see text] implies [Formula: see text] for any [Formula: see text]. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more ge...

A submodule $W$ of $V$ is summand absorbing, if $x + y \in W$ implies $x \in W, \; y \in W $ for any $x, y \in V$. Such submodules often appear in modules over (additively) idempotent semirings, particularly in tropical algebra. This paper studies amalgamation and extensions of these submodules, and more generally of upper bound modules.

An [Formula: see text]-module [Formula: see text] over a semiring [Formula: see text] lacks zero sums (LZS) if [Formula: see text] implies [Formula: see text]. More generally, a submodule [Formula: see text] of [Formula: see text] is “summand absorbing” (SA), if, for all [Formula: see text], [Formula: see text] These relate to tropical algebra and...

Rays are classes of an equivalence relation on a module V over a supertropical semiring. They provide a version of convex geometry, supported by a "supertropical trigonometry" and compatible with quasilinearity, in which the CS-ratio takes the role of the Cauchy-Schwarz inequality. CS-functions which emerge from the CS-ratio are a useful tool that...

The category $\operatorname{STROP}$ of commutative semirings, whose morphisms are transmissions, is a full and reflective subcategory of the category $\operatorname{STROP}_m$ of supertropical monoids. Equivalence relations on supertropical monoids are constructed easily, and utilized effectively for supertropical semirings, whereas ideals are too s...

Relying on rays, we search for submodules of a module V over a supertropical semiring on which a given anisotropic quadratic form is quasilinear. Rays are classes of a certain equivalence relation on V, that carry a notion of convexity, which is consistent with quasilinearity. A criterion for quasilinearity is specified by a Cauchy-Schwartz ratio w...

This paper is a sequel of [Z. Izhakian, M. Knebusch and L. Rowen, Supertropical semirings and supervaluations, J. Pure Appl. Algebra 220(1) (2016) 61–93], where we introduced quadratic forms on a module [Formula: see text] over a supertropical semiring [Formula: see text] and analyzed the set of bilinear companions of a quadratic form [Formula: see...

Relying on rays, we search for submodules of a module V over a supertropical semiring on which a given anisotropic quadratic form is quasilinear. Rays are classes of a certain equivalence relation on V, that carry a notion of convexity, which is consistent with quasilinearity. A criterion for quasilinearity is specified by a Cauchy-Schwartz ratio w...

An $R$-module $V$ over a semiring $R$ lacks zero sums (LZS) if $ x + y = 0 \; \Rightarrow \; x = y = 0$. More generally, a submodule $W$ of $V$ is summand absorbing in $V$ if $ \forall \, x, y \in V: \ x + y \in W \; \Rightarrow \; x \in W, \; y \in W. $ These arise in tropical algebra and modules over idempotent semirings. We explore the lattice o...

This paper is a sequel to [6], in which we introduced quadratic forms on a module over a supertropical semiring _R_ and analyzed the set of bilinear companions of a single quadratic form in case the module _V_ is free. Any (semi)module over a semiring gives rise to what we call its minimal ordering, which is a partial order iff the semiring is “upp...

A direct sum decomposition theory is developed for direct summands (and complements) of modules over a semiring $R$, having the property that $v+w = 0$ implies $v = 0$ and $w = 0$. Although this never occurs when $R$ is a ring, it always does holds for free modules over the max-plus semiring and related semirings. In such situations, the direct com...

We study quadratic forms on free modules with unique base, the situation that arises in tropical algebra, and prove the analog of Witt's Cancellation Theorem. Also, the tensor product of an indecomposable bilinear module $(U, \gamma)$ with an indecomposable quadratic module $(V,q) $ is indecomposable, with the exception of one case, where two indec...

We aim at analyzing a Prüfer extension A ⊂ R in terms of the set S(R∕A) of nontrivial PM-valuations v on R over A (i.e. with A ⊂ A v ), called the restricted PM-spectrum of R over A, in order to understand the lattice of overrings of A in R. We engage S(R∕A) as partially ordered set (v ≤ w iff A v ⊂ A w ), although it would be more comprehensive to...

The all over idea of the present chapter is to associate to any ring extension A ⊂ R a commuting square of ring extensions such that B is Prüfer in T and there exists a process v↦v ∗ which associates with v in a suitable family \(\mathfrak{M}\) of valuations of R over A a special valuation v of T over B such that v ∗∘ j = v. (“Over A” means that A...

We embed the important work of Gräter on approximation theorems in the book. Approximation theorems are a well-known and important topic in classical valuation theory of fields. The question is to decide for given valuations v 1, …, v n of a field, elements a 1, …, a n in the field and α 1, …, α n in the value groups whether there is an element x i...

Supertropical monoids are a structure slightly more general than the supertropical semirings, which have been introduced and used by the first and the third authors for refinements of tropical geometry and matrix theory in [IR1]-[IR3], and then studied by us in a systematic way in [IKR1]-[IKR3] in connection with "supervaluations". In the present p...

We initiate the theory of a quadratic form $q$ over a semiring $R$. As customary, one can write $$q(x+y) = q(x) + q(y)+ b(x,y),$$ where $b$ is a companion bilinear form. But in contrast to the ring-theoretic case, the companion bilinear form need not be uniquely defined. Nevertheless, $q$ can always be written as a sum of quadratic forms $q = \kapp...

Tropical mathematics often is defined over an ordered cancellative monoid $\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted algebraic structure theory, and also do not reflect certain valuation-theoretic properties, thereby forcing research...

This paper supplements the authors’ preprint [Layered tropical mathematics, arxiv:0912.1398], showing that categorically the layered theory is the same as the theory of ordered monoids (e.g. the max-plus algebra) used in tropical mathematics. A layered theory is developed in the context of categories, together with a “tropicalization functor” which...

We generalize the constructions of [17,19] to layered semirings, in order to enrich the structure and provide finite examples for applications in arithmetic (including finite examples). The layered category theory of [19] is extended accordingly, to cover noncancellative monoids.

Continuing 44. Izhakian , Z , Knebusch , M and Rowen , L . Supertropical linear algebra, preprint 2010. Available at arXiv:1008.0025 View all references, this article investigates finer points of supertropical vector spaces, including dual bases and bilinear forms, with supertropical versions of standard classical results such as the Gram–Schmid...

We interpret a valuation v on a ring R as a map v:R→M into a so-called bipotent semiring M (the usual max–plus setting), and then define a supervaluationφ as a suitable map into a supertropical semiring U with ghost ideal M (cf. Izhakian and Rowen (2010, in press) and ) covering v via the ghost map U→M. The set Cov(v) of all supervaluations coverin...

We complement two papers on supertropical valuation theory ([1111. Izhakian , Z. , Knebusch , M. , Rowen , L. ( 2011 ). Supertropical semirings and supervaluations . J. Pure and Applied Alg. 215 ( 10 ): 2431 – 2463 . Preprint at arXiv:1003.1101 .[CrossRef], [Web of Science ®]View all references], [1212. Izhakian , Z. , Knebusch , M. , Rowen ,...

This paper is a sequel of [IKR1], where we defined supervaluations on a commutative ring $R$ and studied a dominance relation $\phi \geq \psi$ between supervaluations $\phi$ and $\psi$ on $R$, aiming at an enrichment of the algebraic tool box for use in tropical geometry. A supervaluation $\phi:R \to U$ is a multiplicative map from $R$ to a supertr...

We give a short tour through major parts of a recent long paper [IKR1] on supertropical valuation theory, leaving aside nearly all proofs (to be found in [IKR1]). In this way we hope to give easy access to ideas of a new branch of so called "supertropical algebra". Comment: 10 pages

We also allow the case dim φ = 0, standing for the unique bilinear form on the zero vector space, the form φ = 0. We agree that the form φ = 0 has good reduction and set λ *(φ) = 0. Problem 1.2. Is this definition meaningful? Up to isometry λ *(φ) should be independent of the choice of the matrix (c ij ). We shall later see that this is indeed the...

From now on we leave the geometric arena behind, and mostly talk of quadratic and bilinear forms, instead of spaces, over fields. The importance of the geometric point of view was to bring quadratic and bilinear modules over valuation rings into the game. For our specialization theory, these modules were merely an aid however, and their rôle has no...

Starting from the map λ W we developed a specialization theory for bilinear forms in §1.3. In particular we associated to a form φ with good reduction with respect to λ (i.e. a form over K that comes from a nondegenerate bilinear module over o) a form λ *(φ) over L such that {λ *(φ)} = λ W ({φ}) and dim λ *(φ) = dim φ. The form λ *(φ) is only deter...

The theory of weak specialization, developed in §1.3, §1.7 and the end of §2.3, has until now played only an auxiliary role, which we could have done without when dealing with quadratic forms (due to Theorem 2.19). For the first time we now come to independent applications of weak specialization.

The objective of this paper is to lay out the algebraic theory of supertropical vector spaces and linear algebra, utilizing the key antisymmetric relation of ``ghost surpasses.''Special attention is paid to the various notions of ``base,'' which include d-base and s-base, and these are compared to other treatments in the tropical theory. Whereas th...

We give a complete list of all quadratic modules and inclusions between them in the ring R[[X]] of formal power series in one variable X over an euclidean field R.

Generalizing supertropical algebras, we present a "layered" structure, "sorted" by a semiring which permits varying ghost layers, and indicate how it is more amenable than the "standard" supertropical construction in factorizations of polynomials, description of varieties, properties of the resultant, and for mathematical analysis and calculus, in...

We outline a specialization theory of quadratic and (symmetric) bilinear forms with respect to a place λ: K → L∪∞. Here K, L denote fields of any characteristic. We have to make a distinction between bilinear forms and quadratic forms and study them both over fields and valuation rings.

Given a commutative ring A equipped with a preordering A+ (in the most general sense, see below), we look for a fractional ring extension (= “ring of quotients” in the sense of Lambek et al. [L]) as big as possible such that A+ extends to a preordering R+ of R (i.e. with A ∩ R+ = A+) in a natural way. We then ask for subextensions A ⊂ B of A ⊂ R su...

We analyse the interplay between real valuations, Prufer extensions and convexity with respect to various preorderings on a given commutative ring. We study all this first in preordered rings in general, then in f-rings. Most often Prufer extensions and real valuations abound whenever a preordering is present. The next logical step, to focus on the...

We analyse the interplay between real valuations, Prüfer extensions and convexity with respect to various preorderings on a given commutative ring. We study all this first in preordered rings in general, then in f-rings. Most often Prüfer extensions and real valuations abound whenever a preordering is present. The next logical step, to focus on the...

Contents. Summary 1 The PM-overrings in a Prüfer extension 2 Regular modules in a PM-extension 3 More ways to characterize PM-extensions, and a look at BM-extensions 4 Tight valuations 5 Existence of various valuation hulls 6 Inside and outside the Manis valuation hull 7 The TV-hull in a valuative extension 8 Principal valuations 9 Descriptions of...

Contents. Summary 1 Multiplicative properties of regular modules 2 Characterizing Prüfer extensions by the behavior of their regular ideals 3 Describing a Prüfer extension by its lattice of regular ideals 4 Tight extensions 5 Distributive submodules 6 Transfer theorems 7 Polars and factors in a Prüfer extension 8 Decomposition of regular modules 9...

Contents. Summary 1 Valuations on rings 2 Valuation subrings and Manis pairs 3 Weakly surjective homomorphisms 4 More on weakly surjective extensions 5 Basic theory of Prüfer extensions 6 Examples of Prüfer extensions and convenient ring extensions 7 Principal ideal results

Appendix A (to I, §4 and I, §5): Flat epimorphisms Appendix B (to II, §2): Arithmetical rings Appendix C (to III, §6): A direct proof of the existence of Manis valuation hulls

. This manuscript describes how a generic splitting tower of a regular anisotropic quadratic form digests the form down to a form which is totally split. Introduction We work with quadratic forms on finite dimensional vector spaces over an arbitrary field k. We call such a form q: V # k regular if the radical V # of the associated bilinear form B q...

Definition 1. Ist (M, S) eine (partiell) geordnete Menge und X ⊆ M eine Teilmenge, so heißt X konvex in M, wenn für alle x, y, z ∈ M gilt: $$x \leqslant z \leqslant y\,und\,x,y \in X \Rightarrow z \in X.

Das zentrale Objekt in diesem Kapitel ist das reelle Spektrum eines Rings, wie es um 1979 von M.F. Roy und M. Coste gefunden wurde, mit seinen elementaren Eigenschaften. Dabei arbeiten wir meist mit ganz allgemeinen kommutativen Ringen. Jedoch ergeben sich in der „geometrischen Situation“, also für die Koordinatenringe affiner Varietäten über reell...

Sei K ein Körper. Definition 1. Eine Anordnung (engl.: ordering) von K ist eine Teilmenge P von K, welche $$\begin{array}{*{20}c} {\left( 1 \right)} \hfill & {P + P \subseteq P,PP \subseteq P,^1 } \hfill \\ {\left( 2 \right)} \hfill & {P \cap \left( { - P} \right) = \left\{ 0 \right\},} \hfill \\ {\left( 3 \right)} \hfill & {P \cup \left( { - P} \r...

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An algebraic proof is given for the following theorem: Every system of n odd polynomials in n + 1 variables over a real closed field R has a common zero on the unit sphere Sn(R) ⊂ Rn+1.

In this chapter we study the ring structure of Witt rings. Starting with a theorem of Witt, which describes the Witt ring of a field F as a quotient of the group ring ℤ[Q(F)], where Q(F) denotes the group F*/F*2 of square classes of the field F, we deduce the structure theorems purely ring-theoretically. Thus the results obtained by this way apply...

Let H be an abstract Wittring with R ≠Rt. We know by theorem 2.7, that the ℤ-valued homomorphisms of R correspond one-one to the minimal prime ideals of R. Since P ⋂ ℤ = 0 for all minimal prime ideals P of R, the minimal prime ideals correspond one-one to the prime ideals of the localisation of R at ℤ {O}, and thus to the elements of Spec \( (\math...

In this introductory chapter we give a brief account of some aspects in the classical theory of symmetric bilinear spaces over fields. For a more detailed discussion the reader should consult [12], [49], [59], [41]. We will often restrict ourselves to fields with characteristic different from two. Remarks concerning peculiarities of the characteris...

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