arXiv subjects

Joel Nagloo

Publications and source records attributed to Joel Nagloo.

17 recordsLinked to original sources

arXiv2026

Special classes of functions

Using model theory and differential algebra, we give necessary conditions for algebraic ordinary differential equations to have a complex Pfaffian solution on some complex domain. These tools also allow us to give many examples of algebraic ordinary differential equations that do not have real Pfaffian solution on any open interval. We also give a sufficient condition for a function to be d-irreducible, in the sense of Nishioka. These characterizations are used to give several answers to questions of Bianconi (2016) and strengthen a theorem of Nguyen (2009).

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arXiv2026

On the number of independent solutions of algebraic differential equations

We prove a conjecture of Kumbhakar, Roy, and Srinivasan (2024) on the classification of order one differential equations, and a conjecture of Kumbhakar and Srinivasan (2025) on higher order equations. Both conjectures involve bounds for the number of independent solutions of the equation and are shown to be results of recent work in differential Galois theory using model theoretic techniques. In both cases, stronger versions of the conjectures hold when working over the field of constants (i.e., when the equation is autonomous). We then use inverse Galois theory to show that the bounds in the conjectures are optimal when working over a differential field which is differentially finitely generated over its constant subfield. We also show how recent results of Jaoui and Moosa (2024) on abelian reductions of differential equations can be used to recover some of the work of Kumbhakar and Srinivasan (2025).

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arXiv2026

Categoricity and non-arithmetic Fuchsian groups

Let $\Gamma \subset PSL_2(\mathbb{R})$ be a non-arithmetic Fuchsian group of the first kind with finite covolume, and let $j_{\Gamma}$ be a corresponding uniformizer. In this paper we introduce a natural $L_{\omega_1,\omega}$-axiomatization $T^{\infty}_{SF}$ of the theory of $j_{\Gamma}$ viewed as a covering map. We show that $T^{\infty}_{SF}$ is categorical in all infinite cardinalities, extending to the non-arithmetic setting earlier results of Daw and Harris obtained in the arithmetic case. We also show that the associated first-order theory $T_{j_{\Gamma}}$ is complete, admits elimination of quantifiers, and is $\omega$-stable.

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arXiv2025

Algebraic independence of the solutions of the classical Lotka-Volterra system

Let $(x_1,y_1),\ldots,(x_n,y_n)$ be distinct non-constant and non-degenerate solutions of the classical Lotka-Volterra system \begin{equation}\notag \begin{split} x'&= axy + bx\\ y'&= cxy + dy, \end{split} \end{equation} where $a,b,c,d\in\mathbb{C}\setminus\{0\}$. We show that if $d$ and $b$ are linearly independent over $\mathbb{Q}$, then the solutions are algebraically independent over $\mathbb{C}$, that is $tr.deg_{\mathbb{C}}\mathbb{C}(x_1,y_1,\ldots,x_n,y_n)=2n$. As a main part of the proof, we show that the set defined by the system in universal differential fields, with $d$ and $b$ linearly independent over $\mathbb{Q}$, is strongly minimal and geometrically trivial. Our techniques also allows us to obtain partial results for some of the more general $2d$-Lotka-Volterra system.

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arXiv2022

Algebraic relations between solutions of Painlevé equations

In this manuscript we make major progress classifying algebraic relations between solutions of Painlevé equations. Our main contribution is to establish the algebraic independence of solutions of various pairs of equations in the Painlevé families; for generic coefficients, we show all algebraic relations between solutions of equations in the same Painlevé family come from classically studied B{ä}cklund transformations. We also apply our analysis of ranks to establish some transcendence results for pairs of Painlevé equations from different families. In that area, we answer several open questions of Nagloo (2016), and in the process answer a question of Boalch (2012). We calculate model theoretic ranks of all Painlevé equations in this article, extending results of Nagloo and Pillay (2017). We show that the type of the generic solution of any equation in the second Painlevé family is geometrically trivial, extending a result of Nagloo (2015). We give the first model theoretic analysis of several special families of the third Painlevé equation, proving results analogous to Nagloo and Pillay (2017). We also give a novel new proof of the irreducibility of the third, fifth and sixth Painlevé equations using recent work of Freitag, Jaoui, and Moosa (2022). Our proof is fundamentally different than the existing transcendence proofs of Watanabe (1998) or Cantat and Loray (2009).

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arXiv2022

Strong minimality of triangle functions

In this manuscript, we give a new proof of strong minimality of certain automorphic functions, originally results of Freitag and Scanlon (2017), Casale, Freitag, and Nagloo (2020), Blázquez-Sanz, Casale, Freitag, and Nagloo (2020). Our proof is shorter and conceptually different than those presently in the literature.

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arXiv2022

On the equations of Poizat and Liénard

We study the structure of the solution sets in universal differential fields of certain differential equations of order two, the Poizat equations, which are particular cases of Liénard equations. We give a necessary and sufficient condition for strong minimality for equations in this class and a complete classification of the algebraic relations for solutions of strongly minimal Poizat equations. We also give an analysis of the non strongly minimal cases as well as applications concerning the Liouvillian and Pfaffian solutions of some Liénard equations.

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arXiv2021

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.

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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.

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arXiv2020

Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

We prove the Ax-Lindemann-Weierstrass theorem with derivatives for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory, monodromy of linear differential equations, the study of algebraic and Liouvillian solutions, differential algebraic work of Nishioka towards the Painlevé irreducibility of certain Schwarzian equations, and considerable machinery from the model theory of differentially closed fields. Our techniques allow for certain generalizations of the Ax-Lindemann-Weierstrass theorem which have interesting consequences. In particular, we apply our results to answer a question of Painlevé (1895). We also answer certain cases of the André-Pink conjecture, namely in the case of orbits of commensurators of Fuchsian groups.

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arXiv2018

Commuting planar polynomial vector fields for conservative Newton systems

We study the problem of characterizing polynomial vector fields that commute with a given polynomial vector field on a plane. It is a classical result that one can write down solution formulas for an ODE that corresponds to a planar vector field that possesses a linearly independent commuting vector field. This problem is also central to the question of linearizability of vector fields. Let $f \in K[x]$, where $K$ is a field of characteristic zero, and $d$ the derivation that corresponds to the differential equation $\ddot x = f(x)$ in a standard way. Let also $H$ be the Hamiltonian polynomial for $d$, that is $H=\frac{1}{2}y^2-\int{f(x)dx}$. It is known that the set of all polynomial derivations that commute with $d$ forms a $K[H]$-module $M_d$. In this paper, we show that, for every such $d$, the module $M_d$ is of rank $1$ if and only if $\text{deg}\; f\geqslant 2$. For example, the classical elliptic equation $\ddot x = 6x^2+a$, where $a \in \mathbb{C}$, falls into this category.

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arXiv2017

Algebraic Independence of generic Painlev\'e Transcendents: P_III and P_VI

We prove that if y"=f(y,y',t) is a generic Painlev\'e equation from the class III and VI, and if y_1,...,y_n are distinct solutions, then y_1,y_1',...,y_n,y_n' are algebraically independent over C(t). This improves the weaker results obtained by the author and Pillay and completely prove the algebraic independence conjecture for the generic Painlev\'e transcendents. In the process, we also prove that any three distinct solutions of a Riccati equation are algebraic independent over C(t), provided that there are no solutions in the algebraic closure of C(t). This answers a very natural question in the theory.

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arXiv2015

On parameterized differential Galois extensions

We prove some existence results on parameterized strongly normal extensions for logarithmic equations. We generalize a result in [Wibmer, Existence of d-parameterized Picard-Vessiot extensions over fields with algebraically closed constants, J. Algebra, 361, 2012]. We also consider an extension of the results in [Kamensky and Pillay, Interpretations and differential Galois extensions, Preprint 2014] from the ODE case to the parameterized PDE case.

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arXiv2015

On Transformations in the Painlevé Family

In this paper we show that generic Painlevé equations from different families are orthogonal. In particular, this means that there are no general Backlund transformations between Painlevé equations from the different families $P_I-P_{VI}$ .

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arXiv2013

Geometric Triviality of the Strongly Minimal Second Painlevé equations

We show that the strongly minimal second Painlevé equation (y" = 2y^3+ty+α) is geometrically trivial, that is we show that if y_1,...,y_n are distinct solutions such that y_1,y_1',y_2,y_2',...,y_n,y_n' are algebraically dependent over C(t), then already for some i<j, y_i,y_i',y_j,y_j' are algebraically dependent over C(t). This gives an extension of some recent result for the second Painlevé equation to the non generic parameters.

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arXiv2012

A Note on Integrability and Internality in DCF0

We investigate the relationship between algebraic integrability and the model theoretic notion of internality. Our main result give a geometric account of almost internality and indeed we show that this notion correspond in a reasonable way to having enough "good" first intergrals.

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arXiv2012

On the algebraic independence of generic Painleve transcendents

We prove that if y" = f(y,y',t) is a generic Painleve equation from among the classes II to V then any collection of distinct solutions and their derivatives are algebraically independent over C(t). (Already proved by Nishioka for the single Painleve I equation). For generic Painleve VI we prove a slightly weaker statement.

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