Séminaires : Structures algébriques ordonnées

In this talk we shall introduce the notion of a new category called the category of superpowersets. Examples superpowersets are found among powersets of nonempty sets, presentable posets in the theory of quadratic forms, and fuzzy subsets of a given set, and we shall see that the category of superpowersets forms a topos. Superpowersets can be equipped with a form of binary operation yielding objects that we shall call superpowergroups. Examples of superpowergroups are naturally built from groups, hypergroups, presentable groups, fuzzy groups and fuzzy hypergroups. The question of when superpowergroups form a topos will be also addressed.

Soit $K$ un corps de caractéristique nulle et $x=(x_1,...,x_r)$. Nous considérons la clôture algébrique de $K[[x]]$ en tant que sous-corps du corps des "séries polyédrales rationnelles" (lui-même sous-corps des séries de Puiseux itérées), et appelons "séries de Puiseux algébroïdes" ses éléments. Nous traitons les deux problèmes suivants :

- étant donné une équation $P(x, y) = 0$ avec $P ∈ K[[x]][y]$, fournir une formule close pour les coefficients d'une série algébroïde solution $y(x)$ en fonction des coefficients de $P$ ;

- étant donné une série algébroïde $y(x)$, reconstruire algorithmiquement les coefficients d'un polynôme annulateur.

Notre stratégie s'appuie sur le traitement de ces deux problèmes à propos des "séries de Puiseux algébriques, c'est-à-dire les éléments de la clôture algébrique de $K(x)$.

Il s'agit d'un travail en commun avec Michel Hickel (U. Bordeaux)."

Sebastian Krapp (Universität Konstanz, Allemagne)

(Joint work with Salma Kuhlmann)
Studying the growth properties of definable functions in o-minimal settings, Miller established the following remarkable growth dichotomy: an o-minimal expansion of an ordered field is either power bounded or admits a definable exponential function (see [2]). Going one step further in the hierarchy of growth, Miller's dichotomy result naturally led to the question whether there exist o-minimal expansions of ordered fields that are not exponentially bounded. Recent research activity in this area is therefore motivated by the search for either an o-minimal expansion of an ordered exponential field by a transexponential function that eventually exceeds any iterate of the exponential or, contrarily, for a proof that any o-minimal expansion of an ordered field is already exponentially bounded.
In this talk, I will present our axiomatic approach towards the study of ordered fields equipped with a transexponential function from [1]. Namely, denoting by e an exponential and by T a compatible transexponential, we establish a first-order theory of ordered transexponential fields in which the functional equation $T(x + 1) = e(T(x))$ holds for any positive $x$. While the archimedean models of this theory are readily described, the study of the non-archimedean models leads to a systematic examination of the induced structure on the residue field and the value group under the natural (i.e. the finest non-trivial convex) valuation. Moreover, I will illustrate construction methods for transexponentials on non-archimedean ordered exponential fields.

All relevant valuation theoretic background will be introduced.

[1] L. S. Krapp and S. Kuhlmann, ˜Ordered transexponential fields™, preprint, 2023, arXiv:2305.04607v2.
[2] C. Miller, "A Growth Dichotomy for O-minimal Expansions of Ordered Fields™, Logic: from Foundations to Applications (eds W. Hodges, M. Hyland, C. Steinhorn and J. Truss; Oxford Sci. Publ., Oxford Univ. Press, New York, 1996) 385–399.

The cone of all positive semidefinite (PSD) real forms in n + 1 (n ≥ 1) variables of degree 2d
(d ≥ 1) contains the subcone of all forms that are representable as finite sums of squares (SOS) of
forms of half degree. In 1888, Hilbert showed in a seminal paper that both cones coincide exactly
in the Hilbert cases n + 1 = 2 or 2d = 2 or (n + 1, 2d) = (3, 4).
In this talk, we construct a filtration of intermediate cones between the SOS and PSD cones
along a filtration of varieties containing the Veronese variety and investigate it for proper inclusions
in any non-Hilbert cases. To this end, we establish a sufficient criterion for a given intermediate
cone to coincide with the SOS cone on the one hand. On the other hand, we develop a tool
with which we are able to determine the greatest cone in the filtration containing an a priori
fixed PSD-extremal not-SOS circuit form. This allows us to introduce and prove a refinement of
Hilbert’s 1888 Theorem in three steps. First, we lay out the situation for (n + 1)-ary quartics
(n ≥ 3). Secondly, we extend our findings to (n + 1)-ary sextics (n ≥ 2) and, then, thirdly, to any
non-Hilbert case via a degree-jumping principle.

Recent results about sums of squares in connection with projective algebraic geometry have relied on the convex geometry of spectrahedra to cover the divide between the two subjects. I want to explain some of these recent results and try to make this bridge by convexity concrete. This is based on joint work with Greg Blekherman, Greg Smith, and Mauricio Velasco.

Let $K$ be a number field and $\mathcal{O}_K$ its ring of integers. A famous result by Knebusch asserts that the natural homomorphism of Witt rings $W\mathcal{O}_K \rightarrow W K$ is injective. This, however, fails to be true if we replace $\mathcal{O}_K$ with an arbitrary ring $R$ whose field of fractions is equal to $K$. We shall consider one particular class of such rings here, namely orders, that is subrings $\mathcal{O}$ of $\mathcal{O}_K$ which are also $\mathbbm{Z}$-modules of rank $n = [K :\mathbbm{Q}]$. The case of orders in quadratic number fields is relatively well understood with both examples of natural homomorphisms of Witt rings being injective and not. In this talk we shall take a closer look at orders in cubic number fields. While orders in quadratic number fields are easy to describe and classify, cubic orders are considerably more difficult to handle. Nevertheless, we manage to
exhibit an example of a cubic order $\mathcal{O}$ whose Witt ring $W\mathcal{O}$ naturally embedds into the Witt ring of its field of fractions.

Nous considérons le cas où les C-structures sont denses, définissablement maximales et sans bijection définissable dans leur arbre canonique entre un intervalle borné d'une branche et un intervalle non borné d'une autre branche (« non borné » signifie ici que la borne supérieure de l'intervalle est une feuille). Elles sont également géométriques et non triviales. Cette situation permet de montrer l'existence de bonnes familles de fonctions, de développer une notion de tangente et de dérivation, et enfin de définir (au sens de la théorie des modèles) un groupe infini.

Dans les années 80 un important courant de pensée en théorie des modèles considérait que des propriétés de pure théorie des modèles de certaines structures abstraites révélaient que ces structures étaient en fait des avatars de structures algébriques classiques. L'exemple même de cette tentative de reconstruction est la conjecture de Cherlin-Zilber : une théorie fortement minimale à géométrie non triviale interprète un groupe infini. Si sa géométrie n'est pas modulaire, elle interprète un corps infini. La conjecture s'est révélée fausse, déjà en ce qui concerne l'existence d'un groupe. Des contre-exemples ne sont apparus qu'au prix de la construction des amalgames de Fraïssé-Hrushovski. Pour qu'elle devienne exacte il a fallu introduire un soupçon de topologie, c'est ce qu'ont fait Ehud Hrushovski et Boris Zilber avec ce qu'ils ont appelé les structures de Zariski. 
Les structures o-minimales portent quant à elles une topologie définissable. Elles sont de plus géométriques, au sens où leur clôture algébrique satisfait le lemme de l'échange. Kobi Peterzil et Sergei Starchenko ont montré qu'elles satisfont (à quelques nuances près) la conjecture de Cherlin-Zilber. 
De façon analogue Fares Maalouf a montré que toute structure C-minimale géométrique modulaire et non triviale permet de définir un groupe. Fares, Patrick Simonetta et moi-même nous intéressons maintenant au cas non modulaire.

(Joint work with Kaique M.A. Roberto (IME-USP) and Hugo R.O. Ribeiro (IME-USP))

A superring is essentially a ring with multivalued sum and product: for instance, the set of polynomials with coefficients in a multiring has a natural structure of superring.
In the first part of the talk, we consider some constructions of superrings, explore some properties of the superring of polynomials with coefficients in a hyperfield and develop some fragments of the theory of algebraic extension for superfields, such as the addition of roots to a superfield.
The significance of these multialgebraic methods to (univalent) Commutative Algebra is indicated by applying these results to algebraic theory of quadratic forms:
(i) obtaining new relevant constructions in the category of special groups (or its equivalent category special hyperfields);
(ii) extending the validity of the Arason-Pfister Hauptsatz - a positive answer to a question posed by Milnor in a classical paper of 1970 - and established by Dickmann-Miraglia in the realm of reduced special groups (or its equivalent category real reduced hyperfields) in 2000.
In the second part of the talk, we expand a fundamental tool in algebraic theory of quadratic forms to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory and Special Groups K-theory, developed by Dickmann-Miraglia. We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in order to provide a solution of Marshall's Signature Conjecture. Moreover, the extended version of Arason-Pfister Hauptsatz entails some interesting properties concerning K-theory and Marshall's conjecture.
We finish this series of talks indicating some future developments of superrings theory and possible applications.

(Joint work with Kaique M.A. Roberto (IME-USP) and Hugo R.O. Ribeiro (IME-USP))
In the first talk of this series, we have presented the concept of multiring and showed how it can (functorially) encode the abstract theories of quadratic forms of special groups and real semigroups.  In this second talk, we present other notions of multirings R (as hyperbolic multirings and quadratic multirings) and pairs (R,T), where R is a multiring and T is a certain multiplicative subset (as DM-multirings, DP-multirings and quadratic pairs), that seem relevant to algebraic theory of quadratic forms. 

Soit T* la théorie des f-anneaux réduits, projetables, divisible-projetables, sc-réguliers et réels clos sans idempotents minimaux (non nuls). La modèle-complétude de cette théorie a été établie auparavant dans le langage des anneaux réticulés avec une relation radicale (dans le sens de Prestel-Schwartz) déterminée par la classe des idéaux premiers minimaux, et une relation de divisibilité locale. Dans cet exposé je décrirai la théorie universelle de T* dans ce langage, augmenté par la relation de divisibilité globale. L'étude détaillé de cette théorie universelle est le pas préalable à une preuve d'élimination des quantificateurs de T* sur laquelle je travaille actuellement.

In this talk we discuss Positivstellensätze in the general context of a unital, commutative, not necessarily finitely generated, real algebra. We focus on the dual problem, and give an introduction to (real) infinite dimensional moment problems (i.e. when measures are supported on real infinite dimensional spaces). We will focus on the following problem: when can a linear functional on a unital commutative real algebra A be represented as an integral w.r.t. a Radon measure on the real character space X(A) equipped with the Borel σ-algebra generated by the weak topology? Our main idea is to construct X(A) as a projective limit of the character spaces of all finitely generated subalgebras of A, to be able to exploit the classical finite dimensional moment theory in the infinite dimensional case. We thus obtain existence results for representing measures defined on the cylinder σ-algebra on X(A), carried by the projective limit construction. If in addition the well-known Prokhorov (ε-K) condition is fulfilled, then we can solve our problem by extending such representing measures from the cylinder to the Borel σ-algebra on X(A). These results allow us to establish e.g. infinite dimensional analogues of the classical Riesz-Haviland.

The purpose is to begin to generalize the theory developed in [DM] (see below) to diagonal quadratic forms with non zero-divisor coefficients in reduced unitary pre-ordered commutative rings. We present:
1. Horn-geometric axioms entailing the equivalence between ring-theoretic isometry and representation by this class of forms is faithfully coded by the corresponding concepts in an associated reduced special group.
A preordered ring, <A, T>, satisying these axioms is called NT-faithfully quadratic (N for "multiplicative set of non zero-divisors in A").
2. A natural correspondence between rings that satisfy the axioms in (1) above and faithfully quadratic rings in the sense of [DM].
3. We prove that any reduced f-ring satisfies the axioms in (1) and, in fact, any preordered f-ring, <A,T>, where T is a preorder satisfying a certain weak cancelation property, verifies the axioms mentioned in item (1). In particular, this applies to real closed rings and rings of continuous functions (bounded or not) over any completely regular topological space.
Joint work with M. Dickmann (IMJ-PRG) and Hugo Ribeiro (USP).
[DM] M. Dickmann, F. Miraglia "Faithfully Quadratic Rings(Memoirs of the AMS 1128, november 2015).

On introduira les relations de divisibilité locale ainsi que ces propriétés élémentaires.  Par la suite on démontrera que la théorie des f-anneaux réels clos, sc-reguliers, projetables, divisible-projetables ayant la premiere propriété de convexité, sans idempotents minimaux est modèle complète en utilisant la relation de divisibilité locale ainsi que la relation radicale associée au spectre des idéaux premiers minimaux. 

In formally real fields K sums of powers with arbitrary even exponents can be
studied by invoking the compact space M = M (K) of all real places, i.e. certain
maps λ : K → P 1 := R ∪ ∞, and the natural representation K → C(M, P 1 ), a 7→ â
based on the evaluation maps â : λ 7→ λ(a). The real holomorphy H = H(K) is
defined as the subring of K consisting of all elements a admitting a finite evaluation
map â : M → R. One is further led to study the series of representations
S n (H) → C(M, S n ), n ∈ N.
In the case of a real (=formally real) function field F over R this general approach
will be substantiated by relating M and H to the family of all smooth models of
F with a compact real locus X which is a real algebraic set and a C ∞ -manifold.
The investigation of line bundles on those manifolds leads to information on the
invertible ideals of H, recent results on continuous rational maps help to study
the image of S n (H) in C(M, S n ). All these informations can be used to derive
qualitative and quantitative results on the representation of an element as a sum of
powers with a given even exponent. Purely algebraic proofs for results of this type
have not been found so far.