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Charlotte Hardouin

Publications and source records attributed to Charlotte Hardouin.

At least 19 recordsLinked to original sources
arXiv2025

A Galois structure on the orbit of large steps walks in the quadrant

The enumeration of weighted walks in the quarter plane reduces to studying a functional equation with two catalytic variables. When the steps of the walk are small, Bousquet-Mélou and Mishna defined a group called the group of the walk which turned out to be crucial in the classification of the small steps models. In particular, its action on the catalytic variables provides a convenient set of changes of variables in the functional equation. This particular set called the orbit has been generalized to models with arbitrary large steps by Bostan, Bousquet-Mélou and Melczer (BBMM). However, the orbit had till now no underlying group. In this article, we endow the orbit with the action of a Galois group, which extends the notion of the group of the walk to models with large steps. As an application, we look into a general strategy to prove the algebraicity of models with small backwards steps, which uses the fundamental objects that are invariants and decoupling. The group action on the orbit allows us to develop a Galoisian approach to these two notions. Up to the knowledge of the finiteness of the orbit, this gives systematic procedures to test their existence and construct them. Our constructions lead to the first proofs of algebraicity of weighted models with large steps, proving in particular a conjecture of BBMM, and allowing to find new algebraic models with large steps.

math.CO↗
arXiv2025

On the D-finiteness of generating functions counting small steps walks in the quadrant

The enumeration of small steps walks confined to the first quadrant of the plane has attracted a lot of attention over the past fifteen years. The associated generating functions are trivariate formal power series in $x,y,t$ where the parameter $t$ encodes the length of the walk while the variables $x,y$ correspond to the coordinates of its ending point. These functions satisfy a functional equation in two catalytic variables. Bousquet-Mélou and Mishna have associated to any small steps model an algebraic curve called the kernel curve and a group called the group of the walk. These two objects turned out to be central in the classification of small steps models. In a recent work, Dreyfus, Elvey Price, and Raschel prove that the group of the walk is finite if and only if the generating function is $D$-finite, that is, it satisfies a linear differential equation with polynomial coefficients in each of its variables $x,y,t$. In this paper, we show that if the group of the walk is infinite, the generating function doesn't satisfy a linear differential equation in $x,y$ or $t$ over the field $\mathbb{Q}(x,y,t)$. The proof of Dreyfus, Elvey Price, and Raschel is based on some singularity analysis. Here, we propose a new strategy which relies essentially on the aforementioned functional equation and on algebraic arguments. This point of view sheds also a new light on the algebraic nature of the generating functions of small steps models since it relates their $D$-finiteness more directly to some geometric properties of the kernel curve.

math.CO↗
arXiv2025

Simplicity of the automorphism group of fields with operators

We adapt a proof of Lascar in order to show the simplicity of the group of automorphisms fixing pointwise all non-generic elements for a class of uncountable models of suitable theories, encompassing both strongly minimal theories as well as several theories of fields with operators.

math.LO↗
arXiv2024

Hypertranscendence and $q$-difference equations over elliptic functionfields

The differential nature of solutions of linear difference equations over the projective line was recently elucidated. In contrast, little is known about the differential nature of solutions of linear difference equations over elliptic curves. In the present paper, we study power series $f(z)$ with complex coefficients satisfying a linear difference equation over a field of elliptic functions $K$,with respect to the difference operator $ϕf(z)=f(qz)$, $2\le q\in\mathbb{Z}$,arising from an endomorphism of the elliptic curve. Our main theoremsays that such an $f$ satisfies, in addition, a polynomial differentialequation with coefficients from $K,$ if and only if it belongs tothe ring $S=K[z,z^{-1},ζ(z,Λ)]$ generated over $K$ by$z,z^{-1}$ and the Weierstrass $ζ$-function. This is the first elliptic extension of recent theorems of Adamczewski, Dreyfus and Hardouin concerning the differential transcendence of solutions of difference equations with coefficients in $\mathbb{C}(z),$ in which various difference operators were considered (shifts, $q$-differenceoperators or Mahler operators). While the general approach, of usingparametrized Picard-Vessiot theory, is similar, many features, andin particular the emergence of monodromy considerations and the ring$S$, are unique to the elliptic case and are responsible for non-trivial difficulties. We emphasize that, among the intermediate results, we prove an integrability result for difference-differential systems over ellipticcurves which is a genus one analogue of the integrability results obtained by Sch\''afke and Singer over the projective line.

math.NT↗
arXiv2024

Galoisian structure of large steps walks in the quadrant

The enumeration of walks in the quarter plane confined in the first quadrant has attracted a lot of attention over the past fifteenth years. The generating functions associated to small steps models satisfy a functional equation in two catalytic variables. For such models, Bousquet-Mélou and Mishna defined a group called the group of the walk which turned out to be central in the classification of small steps models. In particular, its action on the catalytic variables yields a set of change of variables compatible with the structure of the functional equation. This particular set called the orbit has been generalized to models with arbitrary large steps by Bostan, Bousquet-Mélou and Melczer. However, the orbit had till now no underlying group. In this article, we endow the orbit with the action of a Galois group, which extends the group of the walk to models with large steps. Within this Galoisian framework, we generalized the notions of invariants and decoupling. This enable us to develop a general strategy to prove the algebraicity of models with small backward steps. Our constructions lead to the first proofs of algebraicity of weighted models with large steps, proving in particular a conjecture of Bostan, Bousquet-Mélou and Melczer, and allowing us to find new algebraic models with large steps.

math.CO↗
arXiv2020

Functional relations of solutions of q-difference equations

In this paper, we study the algebraic relations satisfied by the solutions of $q$-difference equations and their transforms with respect to an auxiliary operator. Our main tool is the parametrized Galois theories developed in two papers. The first part of this paper is concerned with the case where the auxiliary operator is a derivation, whereas the second part deals a $\mathbf{q'}$-difference operator. In both cases, we give criteria to guaranty the algebraic independence of a series, solution of a $q$-difference equation, with either its successive derivatives or its $\mathbf{q'}$-transforms. We apply our results to $q$-hypergeometric series.

math.NT↗
arXiv2020

On the Kernel curves associated with walks in the quarter plane

The kernel method is an essential tool for the study of generating series of walks in the quarter plane. This method involves equating to zero a certain polynomial, the kernel polynomial, and using properties of the curve, the kernel curve, this defines. In the present paper, we investigate the basic properties of the kernel curve (irreducibility, singularities, genus, uniformization, etc).

math.CO↗
arXiv2020

Length derivative of the generating series of walks confined in the quarter plane

In the present paper, we use difference Galois theory to study the nature of the generating function counting walks with small steps in the quarter plane. These series are trivariate formal power series $Q(x,y,t)$ that count the number of walks confined in the first quadrant of the plane with a fixed set of admissible steps, called the model of the walk. While the variables $x$ and $y$ are associated to the ending point of the path, the variable $t$ encodes its length. In this paper, we prove that in the unweighted case, $Q(x,y,t)$ satisfies an algebraic differential relation with respect to $t$ if and only if it satisfies an algebraic differential relation with respect $x$ (resp. $y$). Combined with other papers, we are able to characterize the $t$-differential transcendence of the $79$ models of walks listed by Bousquet-Mélou and Mishna.

math.CO↗
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

Algebraic independence and linear difference equations

We consider pairs of automorphisms $(ϕ,σ)$ acting on fields of Laurent or Puiseux series: pairs of shift operators $(ϕ\colon x\mapsto x+h_1, σ\colon x\mapsto x+h_2)$, of $q$-difference operators $(ϕ\colon x\mapsto q_1x,\ σ\colon x\mapsto q_2x)$, and of Mahler operators $(ϕ\colon x\mapsto x^{p_1},\ σ\colon x\mapsto x^{p_2})$. Given a solution $f$ to a linear $ϕ$-equation and a solution $g$ to a linear $σ$-equation, both transcendental, we show that $f$ and $g$ are algebraically independent over the field of rational functions, assuming that the corresponding parameters are sufficiently independent. As a consequence, we settle a conjecture about Mahler functions put forward by Loxton and van der Poorten in 1987. We also give an application to the algebraic independence of $q$-hypergeometric functions. Our approach provides a general strategy to study this kind of question and is based on a suitable Galois theory: the $σ$-Galois theory of linear $ϕ$-equations.

math.NT↗
arXiv2020

On Differentially Algebraic Generating Series for Walks in the Quarter Plane

We refine necessary and sufficient conditions for the generating series of a weighted model of a quarter plane walk to be differentially algebraic. In addition, we give algorithms based on the theory of Mordell-Weil lattices, that, for each weighted model, yield polynomial conditions on the weights determining this property of the associated generating series.

math.CO↗
arXiv2020

Hypertranscendence and linear difference equations

After Hölder proved his classical theorem about the Gamma function, there has been a whole bunch of results showing that solutions to linear difference equations tend to be hypertranscendental i.e. they cannot be solution to an algebraic differential equation). In this paper, we obtain the first complete results for solutions to general linear difference equations associated with the shift operator $x\mapsto x+h$ ($h\in\mathbb{C}^*$), the $q$-difference operator $x\mapsto qx$ ($q\in\mathbb{C}^*$ not a root of unity), and the Mahler operator $x\mapsto x^p$ ($p\geq 2$ integer). The only restriction is that we constrain our solutions to be expressed as (possibly ramified) Laurent series in the variable $x$ with complex coefficients (or in the variable $1/x$ in some special case associated with the shift operator). Our proof is based on the parametrized difference Galois theory initiated by Hardouin and Singer. We also deduce from our main result a general statement about algebraic independence of values of Mahler functions and their derivatives at algebraic points.

math.NT↗
arXiv2020

Walks in the quarter plane, genus zero case

We use Galois theory of difference equations to study the nature of the generating series of (weighted) walks in the quarter plane with genus zero kernel curve. Using this approach, we prove that the generating series do not satisfy any nontrivial (possibly nonlinear) algebraic differential equation with rational coefficients.

math.CO↗
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↗
arXiv2019

Hypertranscendence of solutions of Mahler equations

The last years have seen a growing interest from mathematicians in Mahler functions. This class of functions includes the generating series of the automatic sequences. The present paper is concerned with the following problem, which is omnipresent in combinatorics: a set of Mahler functions $u_{1},...,u_{n}$ being given, are $u_{1},...,u_{n}$ and their successive derivatives algebraically independent? In this paper, we give general criteria ensuring an affirmative answer to this question. We apply our main results to the generating series attached to the so-called Baum-Sweet and Rudin-Shapiro automatic sequences. In particular, we show that these series are hyperalgebraically independent, i.e., that these series and their successive derivatives are algebraically independent. Our approach relies of the parametrized difference Galois theory (in this context, the algebro-differential relations between the solutions of a given Mahler equation are reflected by a linear differential algebraic group).

math.AC↗
arXiv2018

On the nature of the generating series of walks in the quarter plane

In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, {\it i.e.}, we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational coefficients.

math.CO↗
arXiv2016

Calculating differential Galois groups of parametrized differential equations, with applications to hypertranscendence

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman and M.F. Singer presented an algorithm calculating the differential Galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators. We use their algorithm as a part of our algorithm. As a result, we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter.

math.AC↗