arXiv subjects

David Blázquez-Sanz

Publications and source records attributed to David Blázquez-Sanz.

At least 19 recordsLinked to original sources

arXiv2026

Generalized linear cellular automata in groups and difference Galois theory II

In the first part of this series the $σ$-ring spanned by the periodic solutions of a generalized linear cellular automaton in a discrete group was shown to be Hopf--Galois, with pro-algebraic Galois group, while a Galois theory for the $σ$-ring spanned by the finite support solutions was left open, with parameterized difference Galois theory suggested as the tool. We apply that tool to automata over $\mathbb Z$. The Fourier transform turns the automaton into a rank one equation $σ(\hat x)=\hatα\hat x$ over a field on which $δ=z\,d/dz$ acts, and we compute, and algorithmically decide, its parameterized Galois group: for genuine automata with symbols rational in the time variable it is either $\mathbb{G}_m(\mathbb C)$ or $\mathbb{G}_m$, the dividing line being separability of the symbol, $\hatα(z,t)=γ(z)ψ(t)$. Finally, the parameterized group in the time direction decides holonomy in the space direction; as a corollary, the Stirling numbers of the first kind are not spatially holonomic.

math.DS↗

arXiv2026

On the relations between several notions of symmetry for the second order linear differential equation

There are several non-equivalent notions of infinitesimal symmetry in the literature of second order linear differential equations: Lie point symmetries, vertical (gauge) symmetries, operator symmetries, infinitesimal contact symmetries, and Lie--Bäcklund operators. We construct an explicit correspondence among the first three, We then describe the Lie algebra $\mathcal L_Ω(\mathbb U)$ of infinitesimal contact transformations of the contact system $Ω=\langle dy-y'\,dx\rangle$, with coefficients in a differential field $\U$ of functions of $x$ and $y$. We obtain a canonical decomposition $\mathcal L_Ω(\mathbb U)=\prod_{k\ge0}\mathcal L_Ω^k\mathbb U$ into $\mathbb C$-vector spaces, each parametrized by $\mathbb U$ (by $\mathbb U\oplus\mathbb U$ for $k=0$), and we compute the algebraic differential formulae for the Lie bracket in these coordinates. Applied to the symmetry problem, we prove that a contact vector field with generating function $W$ is a symmetry of if and only if $A^{2}W=aW+b\,AW$, where $A$ is the vector field in the jet space corresponding to the equation; equivalently, if and only if $W=F_1(u_1,u_2)ϕ_1+F_2(u_1,u_2)ϕ_2$ for arbitrary functions $F_1,F_2$ of the two first integrals of $A$ and a basis $ϕ_1,ϕ_2$ of solutions. The symmetry algebra is always parametrized by two arbitrary functions of two variables. It also shows that the decomposition of $\Lom(\U)$ never captures the whole symmetry algebra: for $a\neq0$ the graded part reduces to the point symmetries, while for $y''=0$ it is an infinite dimensional but still \emph{proper} subspace, and in neither case is it a Lie subalgebra. Finally we make precise the transformation law $W\mapstoμ^{-1}(W\circφ)$ for characteristics under a contact transformation with conformal factor $μ$, which governs the transport of evolutionary representatives.

math.CA↗

arXiv2026

A differential approach to Ax-Schanuel, I

In this paper, we prove several Ax-Schanuel type results for uniformizers of geometric structures; our general results describe the differential algebraic relations between the solutions of the partial differential equations satisfied by the uniformizers. In particular, we give a proof of the full Ax-Schanuel Theorem with derivatives for uniformizers of simple projective structure on curves including unifomizers of any Fuchsian group of the first kind and any genus. Combining our techniques with those of Ax, we give a strong Ax-Schanuel result for the combination of the derivatives of the j-function and the exponential function. In the general setting of Shimura varieties, we obtain an Ax-Schanuel theorem for the derivatives of uniformizing maps. Our techniques combine tools from differential geometry, differential algebra and the model theory of differentially closed fields.

math.NT↗

arXiv2023

Minimality of the $\mathcal D$-groupoid of symmetries of a projective structure

In this article we study Kummer's $\mathcal D$-groupoid, which is the groupoid of symmetries of a meromorphic projective structure. We give necessary and sufficient conditions for its minimality, in the sense of not having infinite sub-$\mathcal D$-groupoids. The condition that we find turns out to be equivalent to the strong minimality of the non-linear Schwarzian equation and the non-integrability by means of Liouvillian functions of the linear Schwarzian equation.

math.AG↗

arXiv2021

Malgrange-Galois groupoid of Painlevé VI equation with parameters

The Malgrange-Galois groupoid of Painlevé IV equations is known to be, for very general values of parameters, the pseudogroup of transformations of the phase space preserving a volume form, a time form and the equation. Here we compute the Malgrange-Galois groupoid of Painlevé VI family including all parameters as new dependent variables. We conclude it is the pseoudogroup of transformations preserving parameter values, the differential of the independent variable, a volume form in the dependent variables and the equation. This implies that a solution of Painlevé VI depending analytically on parameters does not satisfy any new partial differential equation (including derivatives w. r. t. parameters) which is not derived from Painlevé VI.

math.DG↗

arXiv2021

Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient

This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at $0$ and $\infty$ of the Laurent polynomial, is seen as an affine algebraic variety. We proof that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies as the doubly confluent and biconfluent Heun equations, and to the theory of algebraically solvable potentials of Shrödinger equations. Also, as an auxiliary tool, we improve a previously known criterium for second order linear differential equations to admit a polynomial solution.

math.CA↗

arXiv2021

Group bundles and group connections

We consider smooth families of Lie groups (group bundles) and connections that are compatible with the group operation. We characterize the space of group connections on a group bundle as an affine space modeled over the vector space of $1$-forms with values cocycles in the Lie algebra bundle of the aforementioned group bundle. We show that group connections satisfy the Ambrose-Singer theorem and that group bundles can be seen as a particular case of associated bundles realizing group connections as associated connections. We give a construction of the Moduli space of group connections with fixed base and fiber, as an space of representations of the fundamental group of the base.

math.DG↗

arXiv2021

Some functional transcendence results around the Schwarzian differential equation

This paper centers around proving variants of the Ax-Lindemann-Weierstrass (ALW) theorem for analytic functions which satisfy Schwarzian differential equations. In previous work, the authors proved the ALW theorem for the uniformizers of genus zero Fuchsian groups, and in this work, we generalize that result in several ways using a variety of techniques from model theory, galois theory and geometry.

math.NT↗

arXiv2020

A simplified categorical approach to several Galois theories

We discuss the concept of Galois structure and Galois epimorphism in a general setting. Namely, a Galois structure for an epimorphism $π\colon M\to B$ in some category ${\mathcal C}$ is the action of a group object that gives to $M$ the structure of principal homogeneous space in the relative category ${\mathcal C}_B$.

math.DG↗

arXiv2019

Liouvillian solutions for second order linear differential equations with polynomial coefficients

In this paper we present an algebraic study concerning the general second order linear differential equation with polynomial coefficients. By means of Kovacic's algorithm and asymptotic iteration method we find a degree independent algebraic description of the spectral set: the subset, in the parameter space, of Liouiville integrable differential equations. For each fixed degree, we prove that the spectral set is a countable union of non accumulating algebraic varieties. This algebraic description of the spectral set allow us to bound the number of eigenvalues for algebraically quasi-solvable potentials in the Schrödinger equation.

math-ph↗

arXiv2019

Homogeneous surfaces admitting invariant connections

We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces admitting exactly one invariant connection.

math.DG↗

arXiv2017

Parallelisms & Lie Connections

The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. The nature of this transcendence is measured by a Galois group built from the Picard-Vessiot theory of principal connections.

math.DG↗

arXiv2017

Symmetry reduction and soliton-like solutions for the generalized Korteweg-de Vries equation

We analyze the gKdV equation, a generalized version of Korteweg-de Vries with an arbitrary function $f(u)$. In general, for a function $f(u)$ the Lie algebra of symmetries of gKdV is the $2$-dimensional Lie algebra of translations of the plane $xt$. This implies the existence of plane wave solutions. Indeed, for some specific values of $f(u)$ the equation gKdV admits a Lie algebra of symmetries of dimension grater than $2$. We compute the similarity reductions corresponding to these exceptional symmetries. We prove that the gKdV equation has soliton-like solutions under some general assumptions, and we find a closed formula for the plane wave solutions, that are of hyperbolic secant type.

math-ph↗

arXiv2016

Closed and asymptotic formulas for energy of some circulant graphs

We consider circulant graphs G(r,N) where the vertices are the integers modulo N and the neighbours of 0 are {-r,...,-1,1,...,r}. The energy of G(r,N) is a trigonometric sum of N*r terms. For low values of r we compute this sum explicitly. We also study the asymptotics of the energy of G(r,N) for big N. There is a known integral formula for the linear growth coefficient, we find a new expression of the form of a finite trigonometric sum with r terms. As an application we show that in the family G(r,N) for r less or equal than 4 there is a finite number of hyperenergetic graphs. On the other hand, for each r>4 there is at most a finite number of non-hyperenergetic graphs of the form G(r,N). Finally we show that the graph G(r,2r+1) minimizes the energy among all the regular graphs of degree 2r.

math.CO↗

arXiv2016

Infinitesimally homogeneous manifolds with prescribed structure groups

We explore the class of triples (M, nabla, P) where M is a manifold, nabla is an affine connection in M and P is a G-structure in M. Inside this class there are infinitesimally homogeneous manifolds, characterized by having G-constant curvature, torsion and inner torsion. For each matrix Lie group G subgroup of GL(Rn) there is a class of infinitesimally homogeneous manifolds with structure group G. In this paper we characterize the classes of infinitesimally homogeneous manifolds for some specific values of the structure group G including: identity group, finite groups, diagonal group, special linear group, orthogonal group and unitary group.

math.DG↗

arXiv2015

Differential Galois Theory and Lie Symmetries

We study the interplay between the differential Galois group and the Lie algebra of infinitesimal symmetries of systems of linear differential equations. We show that some symmetries can be seen as solutions of a hierarchy of linear differential systems. We show that the existence of rational symmetries constrains the differential Galois group in the system in a way that depends of the Maclaurin series of the symmetry along the zero solution.

math.CA↗

arXiv2014

A note on the relation between joint and differential invariants

We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the context of joint invariants. We explore the relation between joint and differential invariants, and we expose a general method that allow to compute differential invariants from joint invariants.

math.DG↗

arXiv2012

Non-integrability Criterium for Normal Variational Equations around an integrable Subsystem and an example: The Wilbeforce spring-pendulum

In this paper we analyze the non-integrability of the Wilbeforce pendulum by means of Morales-Ramis theory in where is enough to prove that the Galois group of the variational equation is not virtually abelian. We obtain these non-integrability results due to the algebrization of the variational equation falls into a Heun differential equation with four singularities and then we apply Kovacic's algorithm to determine its non-integrability.

math.DS↗