arXiv subjects

Francesco Gallinaro

Publications and source records attributed to Francesco Gallinaro.

11 recordsLinked to original sources

arXiv2026

Projective curves and weak second-order logic

Given an algebraically closed field $K$ of characteristic zero, we study the incidence relation between points and irreducible projective curves, or more precisely the poset of irreducible proper subvarieties of $\mathbb P^2(K)$. Answering a question of Marcus Tressl, we prove that the poset interprets the field, and it is in fact bi-interpretable with the two-sorted structure consisting of the field $K$ and a sort for its finite subsets. In this structure one can define the integers, so the theory is undecidable. When $K$ is the field of complex numbers we can nevertheless obtain a recursive axiomatization modulo the theory of the integers. We also show that the integers are stably embedded and that the poset of irreducible varieties over the complex numbers is not elementarily equivalent to the one over the algebraic numbers.

math.LO↗

arXiv2026

On the elementary theory of the real exponential field

Assuming Schanuel's conjecture, we prove that the complete theory $T_{\exp}$ of the real exponential field is axiomatized by the axioms of definably complete exponential fields satisfying $\exp' = \exp$. This implies the result of Macintyre and Wilkie that, under the same conjecture, $T_{\exp}$ is decidable. Our approach is based on the model completeness of a similar set of axioms for the exponential function restricted to $(-1,1)$, which we prove unconditionally.

math.LO↗

arXiv2026

The Fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations

We prove a fundamental theorem for tropical partial differential equations, analogous to the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al., Falkensteiner et al. and from Fink and Toghani for the case of trivial valuation as introduced by Grigoriev to differential equations with power series coefficients over any valued field. Crucial ingredients are the framework for tropical partial differential equations introduced by Giansiracusa and Mereta and a result on infinite intersections of projections of fibers of tropicalizations, which we prove using Hrushovski and Loeser's model-theoretic interpretation of Berkovich analytification. As a corollary of the fundamental theorem, we show that the radius of convergence of solutions of an ordinary differential equation over a nontrivially valued field can be computed tropically.

math.AG↗

arXiv2025

Automorphisms of valued fields: amalgamation and existential closedness

We study valued fields equipped with an automorphism. We prove that all of them have an extension admitting an equivariant cross-section of the valuation. In residual characteristic zero, and in the presence of such a cross-section, we show that amalgamation problems are solvable precisely when the induced residual problem is, characterise the existentially closed objects of this category, and prove that its positive theory does not have the tree property of the second kind. We prove analogous results with cross-sections replaced by angular components. Along the way, we show that array modelling does not require thickness.

math.LO↗

arXiv2025

Likely intersections in powers of the multiplicative group

We derive two finiteness properties as consequences of the geometrical non-degeneracy of an algebraic subvariety $W$ of a power of the multiplicative group, concerning the intersections of $W$ with translates of a subtorus $H$ of dimension greater than or equal to the codimension of $W$. The first one is that every translate of $H$ intersects $W$, unless $H$ is contained in one of finitely many proper subtori depending only on $W$. The second one is that every translate of $H$ by a torsion point intersects $W$, unless the translate is contained in one of finitely many proper algebraic subgroups, again depending only on $W$. We use methods from tropical geometry and equidistribution, as well as some very mild model theory.

math.NT↗

arXiv2024

Exponential sums equations and the Exponential Closedness conjecture

This is an expository paper aiming to introduce Zilber's Exponential Closedness conjecture to a general audience. Exponential Closedness predicts when (systems of) equations involving addition, multiplication, and exponentiation have solutions in the complex numbers. It is a natural statement at the boundary between complex geometry and algebraic geometry. While it is open in full generality, many special cases and variants have been proven in the last two decades. In the first part of the paper we give a proof of a special case of the conjecture, namely, we show that exponential sums in a single complex variable, such as $e^{\sqrt{2}z}+3i e^{5z}+π$, always have infinitely many zeroes. This follows from well-known results in the literature and can actually be proven by standard complex analytic tools. We present a proof, motivated by the ideas of Zilber and Gallinaro, which uses simplified versions of some ingredients of more advanced techniques in the area, thus allowing the reader to explore these advanced proofs on a simple example. Our approach uses only elementary techniques (from standard undergraduate algebra and complex analysis), and the proofs of some of the main ingredients appear to be new. Moreover, these ingredients come from various areas of mathematics, such as functional transcendence and complex geometry, and are a useful way of introducing some classical concepts in these areas to the reader. In the second part we explain the Exponential Closedness conjecture and discuss some known special cases, not least the so-called ``raising to powers'' case (due to Gallinaro) which generalises the above-mentioned result and whose proof uses advanced versions of the tools presented in the first part of the paper.

math.CV↗

arXiv2023

Solving Systems of Equations of Raising-to-Powers Type

We address special cases of the analogues of the exponential algebraic closedness conjecture relative to the exponential maps of semiabelian varieties and to the modular $j$ function. In particular, we show that the graph of the exponential of an abelian variety intersects products of free rotund varieties in which the subvariety of the domain is a sufficiently generic linear subspace, and that the graph of $j$ intersects products of free broad varieties in which the subvariety of the domain is a Möbius subvariety.

math.LO↗

arXiv2023

Dividing Lines between Positive Theories

We give definitions of the properties OP, IP, $k$-TP, TP$_1$, $k$-TP$_2$, SOP$_1$, SOP$_2$ and SOP$_3$ in positive logic, and prove various implications and equivalences between them. We also provide a characterisation of stability in positive logic in analogy with the one in full first-order logic, both on the level of formulas and on the level of theories. For simple theories there are the classically equivalent definitions of not having TP and dividing having local character, which we prove to be equivalent in positive logic as well. Finally, we show that a thick theory $T$ has OP iff it has IP or SOP$_1$ and that $T$ has TP iff it has SOP$_1$ or TP$_2$, analogous to the well-known results in full first-order logic where SOP$_1$ is replaced by SOP in the former and by TP$_1$ in the latter. Our proofs of these final two theorems are new and make use of Kim-independence.

math.LO↗

arXiv2023

Quasiminimality of complex powers

The complex field, equipped with the multivalued functions of raising to each complex power, is quasiminimal, proving a conjecture of Zilber and providing evidence towards his stronger conjecture that the complex exponential field is quasiminimal.

math.LO↗

arXiv2023

Exponential Sums Equations and Tropical Geometry

We show a case of Zilber's Exponential-Algebraic Closedness Conjecture, establishing that the conjecture holds for varieties which split as the product of a linear subspace of the additive group $\mathbb{C}^n$ and an algebraic subvariety of the multiplicative group $(\mathbb{C}^\times)^n$. This amounts to solving certain systems of exponential sums equations, and it generalizes old results of Zilber, which required stronger assumptions on the variety such as the linear space being defined over the real numbers. The proofs use the theory of amoebas and tropical geometry.

math.LO↗

arXiv2023

On Some Systems of Equations in Abelian Varieties

We solve a case of the Abelian Exponential-Algebraic Closedness Conjecture, a conjecture due to Bays and Kirby, building on work of Zilber, which predicts sufficient conditions for systems of equations involving algebraic operations and the exponential map of an abelian variety to be solvable in the complex numbers. More precisely, we show that the conjecture holds for subvarieties of the tangent bundle of an abelian variety $A$ which split as the product of a linear subspace of the Lie algebra of $A$ and an algebraic variety. This is motivated by work of Zilber and of Bays-Kirby, which establishes that a positive answer to the conjecture would imply quasiminimality of certain structures on the complex numbers. Our proofs use various techniques from homology (duality between cup product and intersection), differential topology (transversality) and o-minimality (definability of Hausdorff limits), hence we have tried to give a self-contained exposition.

math.LO↗