arXiv subjects

Pablo Cubides Kovacsics

Publications and source records attributed to Pablo Cubides Kovacsics.

At least 19 recordsLinked to original sources

arXiv2026

Quantifier elimination for lovely pairs of strongly geometric fields

Let $T$ be a complete strongly geometric theory of fields with quantifier elimination. We show that the theory of lovely pairs of $T$ has quantifier elimination in Delon's definitional expansion by predicates for linear independence and function symbols for the corresponding coordinate functions. Apart from recovering Delon's original results for pairs of algebraically closed fields and dense pairs of algebraically closed valued fields, we obtain as particular cases, quantifier elimination for theories of dense pairs of real closed and $p$-adically closed fields.

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arXiv2024

On residual domination and types orthogonal to the value group

We present a unifying framework of residual domination for (expansions of) henselian valued fields of equicharacteristic zero, encompassing some valued fields with operators. We show that the class of residually dominated types coincides with the types that are orthogonal to the value group, and with the class of types whose reduct to ACVF (the theory of algebraically closed valued fields with a non-trivial valuation) are generically stable. When the residue field is stable (resp. simple) we relate these equivalent notions to generic stability (resp. simplicity). Those results apply in particular to ultraproducts of $p$-adic fields and to the limit theory VFA$_{0}$ of algebraically closed valued fields of characteristic $p$ with the Frobenius automorphism (as $p$ tends to infinity).

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arXiv2024

A Characterization of Quasi-homogeneous Bivariate Polynomials

If a reduced bivariate polynomial is quasi-homogeneous, then its discriminant is a monomial. Over fields of characteristic $0$, we show that if one adds another simple condition, this becomes an equivalence. We also give a third equivalent condition that is stated geometrically.

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arXiv2022

Topological fields with a generic derivation

We study a class of tame $\mathcal{L}$-theories $T$ of topological fields and their $\mathcal{L}_δ$-extension $T_δ^*$ by a generic derivation $δ$. The topological fields under consideration include henselian valued fields of characteristic 0 and real closed fields. We show that the associated expansion by a generic derivation has $\mathcal{L}$-open core (i.e., every $\mathcal{L}_δ$-definable open set is $\mathcal{L}$-definable) and derive both a cell decomposition theorem and a transfer result of elimination of imaginaries. Other tame properties of $T$ such as relative elimination of field sort quantifiers, NIP and distality also transfer to $T_δ^*$. As an application, we derive consequences for the corresponding theories of dense pairs. In particular, we show that the theory of pairs of real closed fields (resp. of $p$-adically closed fields and real closed valued fields) admits a distal expansion. This gives a partial answer to a question of P. Simon.

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arXiv2021

Beautiful pairs

We introduce an abstract framework to study certain classes of stably embedded pairs of models of a complete $\mathcal{L}$-theory $T$, called \textit{beautiful pairs}, which comprises Poizat's belles paires of stable structures and van den Dries-Lewenberg's tame pairs of o-minimal structures. Using an amalgamation construction, we relate several properties of beautiful pairs with properties analogous to properties in Fra\"{i}ss\'e classes. After characterizing beautiful pairs of various theories of ordered abelian groups and valued fields, including the theories of algebraically closed, $p$-adically closed and real closed valued fields, we show an Ax-Kochen-Ershov type result for beautiful pairs of henselian valued fields. As an application, we derive strict pro-definability of particular classes of definable types. When $T$ is one of the theories of valued fields mentioned above, the corresponding classes of types are related to classical geometric spaces and our main result specializes to their strict pro-definability. Most notably, we exhibit the strict pro-definability of a natural space of types associated to Huber's analytification. In this way, we also recover a result of Hrushovski-Loeser on the strict pro-definability of stably dominated types in algebraically closed valued fields, which corresponds to Berkovich's analytification.

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arXiv2021

Arc-wise analytic t-stratifications

We introduce two new notions of stratifications in valued fields: t$^2$-stratifications and arc-wise analytic t-stratifications. We show the existence of arc-wise analytic t-stratifications in algebraically closed valued fields with analytic structure in the sense of R. Cluckers and L. Lipshitz. We prove that arc-wise analytic t-stratifications are t$^2$-stratifications and, moreover, that t$^2$-stratifications are valuative Lipschitz stratifications as defined by the second author and Y. Yin (the latter ones being closely related to Lipschitz stratifications in the sense of Mostowski). Finally, we introduce a combinatorial invariant associated to a t-stratification which we call the critical value function. We explain how the critical value function of arc-wise analytic t-stratifications can be used to formulate programatic conjectural bounds for the Nash-Semple conjecture.

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arXiv2021

Cohomology of algebraic varieties over non-archimedean fields

We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed non-trivially valued non-archimedean field $K$ based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $Γ_\infty$, where $Γ$ denotes the value group of $K$. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $Γ_\infty$. In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the topological cohomology of the analytification of algebraic varieties concerning finiteness and invariance.

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arXiv2020

Definable completeness of $P$-minimal fields and applications

We show that every definable nested family of closed and bounded subsets of a $P$-minimal field $K$ has non-empty intersection. As an application we answer a question of Darnière and Halupczok showing that $P$-minimal fields satisfy the "extreme value property": for every closed and bounded subset $U\subseteq K$ and every interpretable continuous function $f\colon U \to Γ_K$ (where $Γ_K$ denotes the value group), $f(U)$ admits a maximal value. Two further corollaries are obtained as a consequence of their work. The first one shows that every interpretable subset of $K\timesΓ_K^n$ is already interpretable in the language of rings, answering a question of Cluckers and Halupczok. This implies in particular that every $P$-minimal field is polynomially bounded. The second one characterizes those $P$-minimal fields satisfying a classical cell preparation theorem as those having definable Skolem functions, generalizing a result of Mourgues.

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arXiv2020

Pro-definability of spaces of definable types

We show pro-definability of spaces of definable types in various classical complete first order theories, including complete o-minimal theories, Presburger arithmetic, $p$-adically closed fields, real closed and algebraically closed valued fields and closed ordered differential fields. Furthermore, we prove pro-definability of other distinguished subspaces, some of which have an interesting geometric interpretation. Our general strategy consists in showing that definable types are uniformly definable, a property which implies pro-definability using an argument due to E. Hrushovski and F. Loeser. Uniform definability of definable types is finally achieved by studying classes of stably embedded pairs.

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arXiv2019

Definable sets of Berkovich curves

In this article, we functorially associate definable sets to $k$-analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of $X$ and show that they satisfy a bijective relation with the radial subsets of $X$. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of $k$-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser's theorem on iso-definability of curves. However, our approach can also be applied to strictly $k$-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting.

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arXiv2018

Real closed valued fields with analytic structure

We show quantifier elimination theorems for real closed valued fields with separated analytic structure and overconvergent analytic structure in their natural one-sorted languages and deduce that such structures are weakly o-minimal. We also provide a short proof that algebraically closed valued fields with separated analytic structure (in any rank) are $C$-minimal.

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arXiv2018

On valuation independence and defectless extensions of valued fields

In this article we further develop the theory of valuation independence and study its relation with classical notions in valuation theory such as immediate and defectless extensions. We use this general theory to settle two open questions regarding vector space defectless extensions of valued fields. Additionally, we provide a characterization of such extensions within various classes of valued fields, extending results of Françoise Delon.

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arXiv2018

An example of a $P$-minimal structure without definable Skolem functions

We show there are intermediate $P$-minimal structures between the semi-algebraic and sub-analytic languages which do not have definable Skolem functions. As a consequence, by a result of Mourgues, this shows there are $P$-minimal structures which do not admit classical cell decomposition.

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arXiv2018

Exponential-constructible functions in $P$-minimal structures

Exponential-constructible functions are an extension of the class of constructible functions. This extension was formulated by Cluckers-Loeser in the context of semi-algebraic and sub-analytic structures, when they studied stability under integration. In this paper we will present a natural refinement of their definition that allows for stability results to hold within the wider class of P-minimal structures. One of the main technical improvements is that we remove the requirement of definable Skolem functions from the proofs. As a result, we obtain stability in particular for all intermediate structures between the semi-algebraic and the sub-analytic languages.

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arXiv2018

Definable functions in tame expansions of algebraically closed valued fields

In this article we study definable functions in tame expansions of algebraically closed valued fields. For a given definable function we have two types of results: of type (I), which hold at a neighborhood of infinity, and of type (II), which hold locally for all but finitely many points in the domain of the function. In the first part of the article, we show type (I) and (II) results concerning factorizations of definable functions over the value group. As an application, we show that tame expansions of algebraically closed valued fields having value group $\mathbb{Q}$ (like $\mathbb{C}_p$ and $\overline{\mathbb{F}_p}^{alg}(\!(t^\mathbb{Q})\!)$) are polynomially bounded. In the second part, under an additional assumption on the asymptotic behavior of unary definable functions of the value group, we extend these factorizations over the residue multiplicative structure $\mathrm{RV}$. In characteristic 0, we obtain as a corollary that the domain of a definable function $f\colon X\subseteq K\to K$ can be partitioned into sets $F\cup E\cup J$, where $F$ is finite, $f|E$ is locally constant and $f|J$ satisfies locally the Jacobian property.

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arXiv2017

A proof of Liouville's theorem via o-minimality

In this short note we give a proof of Liouville's theorem (every bounded entire complex function is constant) following Peterzil and Starchenko's approach to complex analysis via o-minimality.

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arXiv2017

Strong density of definable types and closed ordered differential fields

The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable subset X of M^n, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.

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arXiv2017

On O-Minimal Expansions of $(\mathbb{Q},<,+,0)$

Let $f:\mathbb{Q}\to \mathbb{Q}$ be a function definable in an o-minimal expansion of $(\mathbb{Q},<,+,0)$. We show that $f$ is eventually linear. In addition, we show that this holds in every elementary equivalent structure.

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