arXiv subjects

Lucia Di Vizio

Publications and source records attributed to Lucia Di Vizio.

At least 19 recordsLinked to original sources

arXiv2026

The Carlitz module and a differential Ax-Lindemann-Weierstrass theorem for the Euler gamma function

We prove a differential transcendence result of type "Ax-Lindemann-Weierstrass" for Euler's gamma function. Given meromorphic functions $ζ_1,\dots,ζ_n$ of a complex variable $ν$ that are pairwise distinct modulo $\mathbb Z$ and algebraic over the field $k$ of meromorphic $1$-periodic functions, the functions $ Γ(ν-ζ_1(ν)),\dots,Γ(ν-ζ_n(ν))$ are differentially independent over the field $k(ν)$. We determine the structure of certain difference field extensions related to the torsion of an avatar of the Carlitz module over meromorphic functions. These extensions are abelian and purely transcendental, the latter property being crucial in our main result, and obtained applying a criterion of differential algebraicity of Hardouin and Singer.

math.NT↗

arXiv2025

Singular walks in the quarter plane and Bernoulli numbers

We consider singular (aka genus $0$) walks in the quarter plane and their associated generating functions $Q(x,y,t)$, which enumerate the walks starting from the origin, of fixed endpoint (encoded by the spatial variables $x$ and $y$) and of fixed length (encoded by the time variable $t$). We first prove that the previous series can be extended up to a universal value of $t$ (in the sense that this holds for all singular models), namely $t=\frac{1}{2}$, and we provide a probabilistic interpretation of $Q(x,y,\frac{1}{2})$. As a second step, we refine earlier results in the literature and show that $Q(x,y,t)$ is indeed differentially transcendental for any $t\in(0,\frac{1}{2}]$. Moreover, we prove that $Q(x,y,\frac{1}{2})$ is strongly differentially transcendental. As a last step, we show that for certain models the series expansion of $Q(x,y,\frac{1}{2})$ is directly related to Bernoulli numbers. This provides a second proof of its strong differential transcendence.

math.CO↗

arXiv2024

Differential transcendence of Bell numbers and relatives: a Galois theoretic approach

In 2003 Klazar proved that the ordinary generating function of the sequence of Bell numbers is differentially transcendental over the field $\mathbb{C}(\{t\})$ of meromorphic functions at $0$. We show that Klazar's result is an instance of a general phenomenon that can be proven in a compact way using difference Galois theory. We present the main principles of this theory in order to prove a general result about differential transcendence over $\mathbb{C}(\{t\})$, that we apply to many other (infinite classes of) examples of generating functions, including as very special cases the ones considered by Klazar. Most of our examples belong to Sheffer's class, well studied notably in umbral calculus. They all bring concrete evidence in support to the Pak-Yeliussizov conjecture, according to which a sequence whose both ordinary and exponential generating functions satisfy nonlinear differential equations with polynomial coefficients necessarily satisfies a linear recurrence with polynomial coefficients.

math.NT↗

arXiv2023

Inhomogeneous order 1 iterative functional equations with applications to combinatorics

We show that if a Laurent series $f\in\mathbb{C}((t))$ satisfies a particular kind of linear iterative equation, then $f$ is either a rational function or it is differentially transcendental over $\mathbb{C}(t)$. This condition is more precisely stated as follows: We consider $R,b\in \mathbb{C}(t)$ with $R(0)=0$, such that $f(R(t))=f(t)+b(t)$. If either $R'(0)=0$ or $R'(0)$ is a root of unity, then either $f$ is a rational function, or $f$ does not satisfy a polynomial differential equation. More generally a solution of a functional equation of the form $f(R(t))=a(t)f(t)+b(t)$ will be either differentially trascendental or the solution of an inhomogeneous linear differential equation of order $1$ with rational coefficients. We illustrate how to apply these results to deduce the differential transcendence of combinatorial generating functions by considering three examples: the ordinary generating function for a family of complete trees; the Green function for excursions on the Sierpinski graph; and a series related to the enumeration of permutations avoiding the consecutive pattern 1423. The proof strategy is inspired by the Galois theory of functional equations and relies on the property of the dynamics of $R$.

math.CO↗

arXiv2021

A Galoisian proof of Ritt theorem on the differential transcendence of Poincaré functions

Using Galois theory of functional equations, we give a new proof of the main result of the paper "Transcendental transcendency of certain functions of Poincaré" by J.F. Ritt, on the differential transcendence of the solutions of the functional equation R(y(t))=y(qt), where R is a rational function with complex coefficients which verifies R(0)=0, R'(0)=q, where q is a complex number with |q|>1. We also give a partial result in the case of an algebraic function R.

math.DS↗

arXiv2020

Difference Galois Theory For The "Applied" Mathematician

The lecture notes below correspond to the course given by the author in occasion of the VIASM school on Number Theory (18-24 June 2018, Hanoi). We have chosen to omit the proofs that are already presented in details in many references in the literature, although they were explained during the lectures, and we have devoted more space to statements useful in the applications, in particular to differential transcendence.

math.NT↗

arXiv2020

Galois theories for $q$-difference equations: comparison theorems

We establish some comparison results among the different parameterized Galois theories for $q$-difference equations, completing the work by CHatzidakis, Hardouin and Singer, that addresses the problem in the case without parameters. Our main result is the link between the abstract parameterized Galois theories, that give information on the differential properties of abstract solutions of $q$-difference equations, and the properties of meromorphic solutions of such equations. Notice that a linear $q$-difference equation with meromorphic coefficients always admits a basis of meromorphic solutions, as proven by Praagman.

math.QA↗

arXiv2020

Reduced Forms of Linear Differential Systems and the Intrinsic Galois-Lie Algebra of Katz

Generalizing the main result of [Aparicio-Monforte A., Compoint E., Weil J.-A., J. Pure Appl. Algebra 217 (2013), 1504-1516], we prove that a linear differential system is in reduced form in the sense of Kolchin and Kovacic if and only if any differential module in an algebraic construction admits a constant basis. Then we derive an explicit version of this statement. We finally deduce some properties of the Lie algebra of Katz's intrinsic Galois group.

math.AG↗

arXiv2019

Intrinsic approach to Galois theory of q-difference equations, with the preface to Part 4 "The Galois D-groupoid of a q-difference system'' by Anne Granier

We give a complete answer to the analogue of Grothendieck conjecture on p-curvatures for q-difference equations defined over K(x), where K is any finitely generated extension of Q and q\in K can be either a transcendental or an algebraic number. This generalizes the results in [DV02], proved under the assumption that K is a number field and q an algebraic number. The results also hold for a field K which is a finite extension of a purely transcendental extension k(q) of a perfect field k. In Part 3, we consider two Galois groups attached to a q-difference module M over K(x): (1) the intrinsic Galois group Gal(M), in the sense of [Kat82]; (2) if char K=0, the intrinsic differential Galois group Gal^D(M), which is a Kolchin differential algebraic group. We deduce an arithmetic description of Gal(M) (resp. Gal^D(M)). In Part 4, we show that the Galois D-groupoid [Gra09] of a nonlinear q-difference system generalizes Gal^D(M).

math.QA↗

arXiv2014

Difference Galois theory of linear differential equations

We develop a Galois theory for linear differential equations equipped with the action of an endomorphism. This theory is aimed at studying the difference algebraic relations among the solutions of a linear differential equation. The Galois groups here are linear difference algebraic groups, i.e., matrix groups defined by algebraic difference equations.

math.AC↗

arXiv2014

Approche galoisienne de la transcendance différentielle

In this survey we present the parameterized Galois theory of difference equations, as introduced by Hardouin-Singer. The purpose of this theory is to give a systematic approach to differential transcendence, also called hypertranscendence. As an example of the applications, we explain the Galoisian proof of Hölder's theorem on the differential transcendence of the Gamma function.

math.CA↗

arXiv2013

Difference algebraic relations among solutions of linear differential equations

We extend and apply the Galois theory of linear differential equations equipped with the action of an endomorphism. The Galois groups in this Galois theory are difference algebraic groups and we use structure theorems for these groups to characterize the possible difference algebraic relations among solutions of linear differential equations. This yields tools to show that certain special functions are difference transcendent. One of our main results is a characterization of discrete integrability of linear differential equations with almost simple usual Galois group, based on a structure theorem for the Zariski dense difference algebraic subgroups of almost simple algebraic groups, which is a schematic version, in characteristic zero, of a result due to Z. Chatzidakis, E. Hrushovski and Y. Peterzil.

math.AC↗

arXiv2012

On the Grothendieck conjecture on p-curvatures for q-difference equations

In the present paper, we give a q-analogue of the Grothendieck conjecture on p-curvatures for q-difference equations defined over the field of rational function K(x), where K is a finite extension of a field of rational functions k(q), with k perfect. Then we consider the generic (also called intrinsic) Galois group in the sense of N. Katz. The result in the first part of the paper lead to a description of the generic Galois group through the properties of the functional equations obtained specializing q on roots of unity. Although no general Galois correspondence holds in this setting, in the case of positive characteristic, where nonreduced groups appear, we can prove some devissage of the generic Galois group. In the last part of the paper, we give a complete answer to the analogue of Grothendieck conjecture on $p$-curvatures for q-difference equations defined over the field of rational function K(x), where K is any finitely generated extension of \mathbb Q and q\neq 0,1: we prove that the generic Galois group of a q-difference module over K(x) always admits an adelic description in the spirit of the Grothendieck-Katz conjecture. To this purpose, if q is an algebraic number, we prove a generalization of the results by L. Di Vizio, 2002.

math.QA↗

arXiv2012

Parameterized generic Galois groups for q-difference equations, followed by the appendix "The Galois D-groupoid of a q-difference system" by Anne Granier

We introduce the parameterized generic Galois group of a q-difference module, that is a differential group in the sense of Kolchin. It is associated to the smallest differential tannakian category generated by the q-difference module, equipped with the forgetful functor. Our previous results on the Grothendieck conjecture for q-difference equations lead to an adelic description of the parameterized generic Galois group, in the spirit of the Grothendieck-Katz's conjecture on p-curvatures. Using this description, we show that the Malgrange-Granier D-groupoid of a nonlinear q-difference system coincides, in the linear case, with the parameterized generic Galois group introduced here. The paper is followed by an appendix by A. Granier, that provides a quick introduction to the D-groupoid of a non-linear q-difference equation.

math.QA↗

arXiv2010

Arithmetic theory of q-difference equations (G_q-functions and q-difference modules of type G, global q-Gevrey series)

In the first part of the paper we give a definition of G_q-function and we establish a regularity result, obtained as a combination of a q-analogue of the Andre'-Chudnovsky Theorem [And89, VI] and Katz Theorem [Kat70, §13]. In the second part of the paper, we combine it with some formal q-analogous Fourier transformations, obtaining a statement on the irrationality of special values of the formal $q$-Borel transformation of a G_q-function.

math.NT↗

arXiv2008

Local analytic classification of $q$-difference equations with $|q|=1$

In this paper, we establish, under convenient diophantine assumptions, a complete analytic classification of $q$-difference modules over the field of germs of meromorphic functions at zero, proving some analytic analogs of the results by Soibelman and Vologodsky, cf. math.AG/0205117, and by Baranovsky and Ginzburg, cf. alg-geom/9607008.

math.QA↗

arXiv2008

Continuity of the radius of convergence of p-adic differential equations on Berkovich analytic spaces

We consider a vector bundle with integrable connection (\cE,\na) on an analytic domain U in the generic fiber \cX_η of a smooth formal p-adic scheme \cX, in the sense of Berkovich. We define the \emph{diameter} δ_{\cX}(ξ,U) of U at ξ\in U, the \emph{radius} ρ_{\cX}(ξ) of the point ξ\in\cX_η, the \emph{radius of convergence} of solutions of (\cE,\na) at ξ, R(ξ) = R_{\cX}(ξ, U,(\cE, \na)). We discuss (semi-) continuity of these functions with respect to the Berkovich topology. In particular, under we prove under certain assumptions that δ_{\cX}(ξ,U), ρ_{\cX}(ξ) and R_ξ(U,\cE,\na) are upper semicontinuous functions of ξ; for Laurent domains in the affine space, δ_{\cX}(-,U) is continuous. In the classical case of an affinoid domain U of the analytic affine line, R is a continuous function.

math.NT↗

arXiv2008

An ultrametric version of the Maillet-Malgrange theorem for nonlinear q-difference equations

We prove an ultrametric q-difference version of the Maillet-Malgrange theorem, on the Gevrey nature of formal solutions of nonlinear analytic q-difference equations. Since °_q and \ord_q define two valuations on {\mathbb C}(q), we obtain, in particular, a result on the growth of the degree in q and the order at q of formal solutions of nonlinear q-difference equations, when q is a parameter. We illustrate the main theorem by considering two examples: a q-deformation of ``Painleve' II'', for the nonlinear situation, and a q-difference equation satisfied by the colored Jones polynomials of the figure 8 knots, in the linear case. We consider also a q-analog of the Maillet-Malgrange theorem, both in the complex and in the ultrametric setting, under the assumption that |q|=1 and a classical diophantine condition.

math.CA↗