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Antongiulio Fornasiero

Publications and source records attributed to Antongiulio Fornasiero.

At least 19 recordsLinked to original sources
arXiv2026

Geometric fields, ranks, and generic derivations

In this note, we show various minimality results for a geometric theory of fields $T$: $T$ is stable if and only if it is strongly minimal, $T$ is simple if and only if it has SU-rank 1, and $T$ is rosy if and only if $T$ is surgical. Combining the first equivalence with an earlier result of Hrushovski, we deduce that algebraically bounded stable fields are precisely expansions of algebraically closed fields by constants. We then consider algebraically bounded and o-minimal expansions of fields with generic derivations. We show that if $\mathbb{M}$ is a simple algebraically bounded structure and $Δ$ is a generic tuple of derivations on $\mathbb{M}$, then $(\mathbb{M};Δ)$ is supersimple if and only if the derivations commute. Similarly, if $\mathbb{M}$ is an o-minimal structure and $Δ$ is a generic tuple of $T$-derivations on $\mathbb{M}$, then $(\mathbb{M};Δ)$ is superrosy if and only if the derivations commute. We obtain explicit bounds on ranks using the Kolchin polynomial.

math.LO↗
arXiv2025

Stable reducts of elementary extensions of Presburger arithmetic

Suppose $N$ is elementarily equivalent to an archimedean ordered abelian group $(G,+,<)$ with small quotients (for all $1 \leq n < ω$, $[G: nG]$ is finite). Then every stable reduct of $N$ which expands $(G,+)$ (equivalently every reduct that does not add new unary definable sets) is interdefinable with $(G,+)$. This extends previous results on stable reducts of $(\mathbb{Z}, +, <)$ to (stable) reducts of elementary extensions of $\mathbb{Z}$. In particular this holds for $G = \mathbb{Z}$ and $G = \mathbb{Q}$. As a result we answer a question of Conant from 2018. This result is a corollary of a more general statement about expansions of weakly-minimal 1-based expansions of abelian groups with small quotients preserving the algebraic closure operator.

math.LO↗
arXiv2025

A weak version of the Strong Exponential Closure

Assuming Schanuel's Conjecture we prove that for any variety V over the algebraic closure over the rational numbers, of dimension n and with dominant projections, there exists a generic point in V. We obtain in this way many instances of the Strong Exponential Closure introduced by Zilber.

math.LO↗
arXiv2024

Hilbert polynomials for finitary matroids

We consider a tuple $Φ= (ϕ_1,\ldots,ϕ_m)$ of commuting maps on a finitary matroid $X$. We show that if $Φ$ satisfies certain conditions, then for any finite set $A\subseteq X$, the rank of $\{ϕ_1^{r_1}\cdotsϕ_m^{r_m}(a):a \in A\text{ and }r_1+\cdots+r_m = t\}$ is eventually a polynomial in $t$ (we also give a multivariate version of the polynomial). This allows us easily recover Khovanskii's theorem on the growth of sumsets, the existence of the classical Hilbert polynomial, and the existence of the Kolchin polynomial. We also prove some new Kolchin polynomial results for differential exponential fields and derivations on o-minimal fields, as well as a new result on the growth of Betti numbers in simplicial complexes.

math.CO↗
arXiv2024

Hilbert polynomial of length functions

Let $λ$ be a general length function for modules over a Noetherian ring R. We use $λ$ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~$λ$. We show that the leading term $μ$ of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for $R[X]$-modules. Similar to algebraic entropy, $μ$ in general is not additive for exact sequence of $R[X]$-modules: we demonstrate how to adapt of certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.

math.AC↗
arXiv2024

Irreducibility in generalized power series

A classical tool in the study of real closed fields are the fields $K((G))$ of generalized power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian group $G$. In this paper we enlarge the family of ordinals $α$ of non-additively principal Cantor degree for which $K((\mathbb{R}^{\le 0}))$ admits irreducibles of order type $α$ far beyond $α=ω^2 $ and $α= ω^3$ known prior to this work.

math.AC↗
arXiv2023

Note on radical and prime E-ideals

We show that the ring of exponential polynomials is not Noetherian even respect to prime E-ideals. Moreover we give a characterization of exponential radical ideals

math.LO↗
arXiv2021

Is the twin prime conjecture independent of Peano Arithmetic?

We show that there is an arithmetical formula F such that ZF proves that F is independent of PA and yet, unlike other arithmetical independent statements, the truth value of F cannot at present be established in ZF or in any other trusted metatheory. In fact we can choose an example of such a formula F such that ZF proves that F is equivalent to the twin prime conjecture.

math.LO↗
arXiv2021

Groups and rings definable in d-minimal structures

We study groups and rings definable in d-minimal expansions of ordered fields. We generalize to such objects some known results from o-minimality. In particular, we prove that we can endow a definable group with a definable topology making it a topological group, and that a definable ring of dimension at least 1 and without zero divisors is a skew field.

math.LO↗
arXiv2021

D-minimal structures

We study d-minimal expansions of ordered fields, and dense pairs thereof. We also consider other generalizations of o-minimality.

math.LO↗
arXiv2020

Generic derivations on o-minimal structures

Let $T$ be a complete, model complete o-minimal theory extending the theory RCF of real closed ordered fields in some appropriate language $L$. We study derivations $δ$ on models $\mathcal{M}\models T$. We introduce the notion of a $T$-derivation: a derivation which is compatible with the $L(\emptyset)$-definable $\mathcal{C}^1$-functions on $\mathcal{M}$. We show that the theory of $T$-models with a $T$-derivation has a model completion $T^δ_G$. The derivation in models $(\mathcal{M},δ)\models T^δ_G$ behaves "generically," it is wildly discontinuous and its kernel is a dense elementary $L$-substructure of $\mathcal{M}$. If $T =$ RCF, then $T^δ_G$ is the theory of closed ordered differential fields (CODF) as introduced by Michael Singer. We are able to recover many of the known facts about CODF in our setting. Among other things, we show that $T^δ_G$ has $T$ as its open core, that $T^δ_G$ is distal, and that $T^δ_G$ eliminates imaginaries. We also show that the theory of $T$-models with finitely many commuting $T$-derivations has a model completion.

math.LO↗
arXiv2020

The addition theorem for locally monotileable monoid actions

We prove an instance of the so-called Addition Theorem for the algebraic entropy of actions of cancellative right amenable monoids $S$ on discrete abelian groups $A$ by endomorphisms, under the hypothesis that $S$ is locally monotileable (that is, $S$ admits a right Følner sequence $(F_n)_{n\in\mathbb N}$ such that $F_n$ is a monotile of $F_{n+1}$ for every $n\in\mathbb N$). We study in details the class of locally monotileable groups, also in relation with already existing notions of monotileability for groups, introduced by Weiss and developed further by other authors recently.

math.GR↗
arXiv2019

Algebraic entropy for amenable semigroup actions

We introduce two notions of algebraic entropy for actions of cancellative right amenable semigroups $S$ on discrete abelian groups $A$ by endomorphisms; these extend the classical algebraic entropy for endomorphisms of abelian groups, corresponding to the case $S=\mathbb N$. We investigate the fundamental properties of the algebraic entropy and compute it in several examples, paying special attention to the case when S is an amenable group. For actions of cancellative right amenable monoids on torsion abelian groups, we prove the so called Addition Theorem. In the same setting, we see that a Bridge Theorem connects the algebraic entropy with the topological entropy of the dual action by means of the Pontryagin duality, so that we derive an Addition Theorem for the topological entropy of actions of cancellative left amenable monoids on totally disconnected compact abelian groups.

math.GR↗
arXiv2017

How to avoid a compact set

A first-order expansion of the $\mathbb{R}$-vector space structure on $\mathbb{R}$ does not define every compact subset of every $\mathbb{R}^n$ if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if $A \subseteq \mathbb{R}^k$ is closed and the Hausdorff dimension of $A$ exceeds the topological dimension of $A$, then every compact subset of every $\mathbb{R}^n$ can be constructed from $A$ using finitely many boolean operations, cartesian products, and linear operations. The same statement fails when Hausdorff dimension is replaced by packing dimension.

math.LO↗
arXiv2013

Towers of complements to valuation rings and truncation closed embeddings of valued fields

We study necessary and sufficient conditions for a valued field $\KF$ with value group $G$ and residue field $\kf$ (with char $\KF$ = char $\kf$) to admit a truncation closed embedding in the field of generalized power series $\kf((G, f))$ (with factor set $f$). We show that this is equivalent to the existence of a family ({\it tower of complements}) of $\kf$-subspaces of $\KF$ which are complements of the (possibly fractional) ideals of the valuation ring. If $\KF$ is a Henselian field of characteristic 0 or, more generally, an algebraically maximal Kaplansky field, we give an intrinsic construction of such a family which does not rely on a given truncation closed embedding. We also show that towers of complements and truncation closed embeddings can be extended from an arbitrary field to at least one of its maximal immediate extensions.

math.AC↗
arXiv2011

A dichotomy for expansions of the real field

A dichotomy for expansions of the real field is established: Either the set of integers is definable or every nonempty bounded nowhere dense definable subset of the real numbers has Minkowski dimension zero.

math.LO↗