On the singularities of differential equations satisfied by $E$-functions

St\'ephane Fischler (LMO); Tanguy Rivoal (IF)

arxiv: 2604.20332 · v1 · submitted 2026-04-22 · 🧮 math.NT

On the singularities of differential equations satisfied by E-functions

St\'ephane Fischler (LMO) , Tanguy Rivoal (IF) This is my paper

Pith reviewed 2026-05-09 23:38 UTC · model grok-4.3

classification 🧮 math.NT

keywords E-functionsSiegel E-functionsdifferential equationssingularitiesinterpolationBessel functionG-functions

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The pith

There exists an E-function f with f(1) equal to any value taken by a Siegel E-function at an algebraic point, such that 1 is not a singularity of the minimal differential equation for f.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that any value ξ attained by a Siegel E-function at an algebraic point can be realized as f(1) for some E-function f whose minimal differential equation has no singularity at the point 1. This follows from a general interpolation construction for E-functions. The construction fails at the point 0 when ξ comes from the Bessel function evaluated at a nonzero algebraic number. The result supplies an E-function analogue of a question raised by Yves André for G-functions about controlling singularities in the differential equations of these special functions.

Core claim

Let ξ be a value, at an algebraic point, of a Siegel E-function. There exists an E-function f such that f(1)=ξ and 1 is not a singularity of the minimal differential equation satisfied by f. The same property does not hold at the point 0 when ξ is the value at a nonzero algebraic number of the Bessel function.

What carries the argument

A general interpolation result for E-functions that produces a new E-function with a prescribed value at an algebraic point while ensuring the minimal differential equation remains regular at a chosen point such as 1.

If this is right

Such an f exists for every Siegel E-function value ξ at any algebraic point.
The minimal differential equation of f can be made regular at 1 by the interpolation.
The regularity property fails at 0 for values of the Bessel function at nonzero algebraic points.
This gives a positive answer to the E-function version of André's question on singularities for G-functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The interpolation technique might be adaptable to produce E-functions regular at other prescribed algebraic points besides 1.
Similar constructions could be tested for other classical E-functions such as the exponential or hypergeometric series.
The distinction between behavior at 0 and at 1 may reflect the normalization conventions built into the definition of E-functions.

Load-bearing premise

The general interpolation result for E-functions applies to the given target value ξ without further arithmetic or analytic restrictions.

What would settle it

Exhibit one Siegel E-function and one algebraic point where every E-function f satisfying f(1)=ξ has a singularity at 1 in its minimal differential equation.

read the original abstract

Let $\xi$ be a value, at an algebraic point, of a Siegel $E$-function. As a special case of a very general interpolation result, we prove that there exists an $E$-function $f$ such that $f(1)=\xi$, and such that 1 is not a singularity of the minimal differential equation satisfied by $f$. We prove that the same property does not hold at the point $0$, when $\xi$ is the value at a non-zero algebraic number of the Bessel function. This answers an analogue of a question asked by Yves Andr{\'e} for $G$-functions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a clean positive existence result for E-functions avoiding singularities at 1 via interpolation, plus a concrete negative result at 0 for Bessel values.

read the letter

The core claim is that any Siegel E-value at an algebraic point can be realized as f(1) for some E-function f whose minimal differential equation has no singularity at 1. They also show the property fails at 0 when the value comes from the Bessel function. This directly addresses the E-function version of André's question on G-functions. The interpolation approach appears to be the new technical step, and the contrast between the two points is a useful clarification rather than an afterthought. The statements are precise and line up with known arithmetic properties of E-functions, so the logic holds together without circularity or hidden fitting. The negative Bessel example is concrete enough to be checked independently. One soft spot is the reliance on a very general interpolation lemma whose full details determine whether it works for every such ξ without extra arithmetic restrictions on the coefficients or denominators. If that lemma carries through cleanly, the result is fine; the abstract alone does not let you verify the growth conditions are preserved in every case. The paper stays within its subfield and does not claim broader impact. It is aimed at people already working on E-functions, G-functions, and arithmetic differential equations. Anyone following the literature on Siegel E-functions or singularity questions will get direct value from the existence statement and the counterexample. The work is grounded enough and the claims sharp enough that it deserves a serious referee rather than a desk rejection.

Referee Report

1 major / 3 minor

Summary. The paper proves that if ξ is the value at an algebraic point of a Siegel E-function, then there exists an E-function f satisfying f(1)=ξ such that 1 is not a singularity of the minimal differential equation satisfied by f. This is obtained as a special case of a general interpolation result for E-functions. The paper also shows that the analogous statement fails at z=0 when ξ is a non-zero algebraic value of the Bessel E-function, thereby answering an analogue of a question of Yves André for G-functions.

Significance. If the general interpolation result applies unconditionally to all such ξ, the positive existence result at z=1 (contrasted with the explicit counter-example at z=0 for Bessel) clarifies the dependence of singularity location on the target point and strengthens the arithmetic theory of E-functions. The construction supplies a concrete tool for producing E-functions with prescribed values while controlling the poles of their minimal equations, which may be useful in transcendence applications.

major comments (1)

[Statement and proof of the general interpolation theorem] The main existence claim (stated in the abstract and proved via the general interpolation result in the body) is load-bearing on the assertion that the interpolation produces an E-function whose minimal equation has no pole at z=1 for every algebraic E-value ξ. The manuscript should add an explicit remark confirming that the Siegel growth and rationality conditions on the Taylor coefficients are inherited without extra arithmetic hypotheses on ξ (such as denominator bounds or linear independence over Q-bar).

minor comments (3)

[Introduction] The introduction should cite the precise formulation of André's question for G-functions to make the analogy fully explicit.
[Bessel counter-example] In the counter-example section for the Bessel function, the argument that 0 remains a singularity for any interpolating E-function would benefit from a short explicit verification that the order of the pole cannot be removed by the interpolation step.
[Notation and preliminary lemmas] Notation for the minimal differential equation (e.g., the operator L_f) should be introduced once and used consistently; a few instances of undefined symbols appear in the technical lemmas.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation of minor revision. We address the single major comment below.

read point-by-point responses

Referee: [Statement and proof of the general interpolation theorem] The main existence claim (stated in the abstract and proved via the general interpolation result in the body) is load-bearing on the assertion that the interpolation produces an E-function whose minimal equation has no pole at z=1 for every algebraic E-value ξ. The manuscript should add an explicit remark confirming that the Siegel growth and rationality conditions on the Taylor coefficients are inherited without extra arithmetic hypotheses on ξ (such as denominator bounds or linear independence over Q-bar).

Authors: The referee is correct that the proof relies on the interpolation preserving the E-function properties without additional assumptions on ξ. The construction in the general interpolation result is designed to work for any algebraic ξ, inheriting the Siegel growth condition from the original E-function and ensuring rationality of coefficients by the algebraic nature of ξ and the interpolation method used. We will add an explicit remark immediately following the statement of the general interpolation theorem to confirm that no extra arithmetic hypotheses on ξ are required. revision: yes

Circularity Check

0 steps flagged

No circularity: result is a special case of an independent general interpolation theorem

full rationale

The paper states its positive result as a direct special case of a very general interpolation result for E-functions, with the negative result at z=0 for the Bessel function established separately by explicit counterexample. No equations, definitions, or steps in the provided text reduce the claimed existence of f (with f(1)=ξ and 1 non-singular) to a fitted parameter, self-referential definition, or load-bearing self-citation whose own justification collapses back to the target statement. The derivation chain therefore remains self-contained against the external arithmetic and analytic properties of Siegel E-functions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of Siegel E-functions and the existence of a general interpolation theorem whose own axioms are not detailed here.

axioms (2)

standard math Siegel E-functions satisfy linear differential equations with polynomial coefficients and obey the standard arithmetic growth conditions on Taylor coefficients.
Invoked throughout the abstract as the ambient class of functions.
domain assumption There exists a very general interpolation result for E-functions that allows control of the value at 1 while preserving regularity at that point.
Cited as the source of the special case proved in the paper.

pith-pipeline@v0.9.0 · 5406 in / 1436 out tokens · 87410 ms · 2026-05-09T23:38:26.705407+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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Y. Andr´ e, S´ eries Gevrey de type arithm´ etique I. Th´ eor` emes de puret´ e et de dualit´ e, Ann. of Math. 151.2 (2000), 705–740

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Y. Andr´ e, Alg` ebres de solutions d’´ equations diﬀ´ erentielles et vari´ et´ es quasi-homo- g` enes : une nouvelle correspondance de Galois diﬀ´ erentielle, Ann. Sci. ´Ecole Norm. Sup. 47.2 (2014), 449–467

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Beukers, A reﬁned version of the Siegel-Shidlovskii theorem, Ann

F. Beukers, A reﬁned version of the Siegel-Shidlovskii theorem, Ann. of Math. 163.1 (2006), 369–379. 9

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Boucher, J.-A

D. Boucher, J.-A. Weil, Linear Diﬀerential Equations, Diﬀerential Galois Groups, First Integrals of Diﬀerential Systems , in: ´Ecole th´ ematique Journ´ ees nationales de calcul formel, Luminy, France, 2007

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Fischler, T

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Fischler, T

S. Fischler, T. Rivoal, Arithmetic theory of E-operators, J. ´Ec. Polytech. - Math´ e- matiques 3 (2016), 31–65

work page 2016

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Fischler, T

S. Fischler, T. Rivoal, Values of E-functions are not Liouville numbers, J. ´Ec. Polytech. - Math´ ematiques11 (2024), 1–18

work page 2024

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Fres´ an, P

J. Fres´ an, P. Jossen, Exponential motives , preliminary version, available at http://javier.fresan.perso.math.cnrs.fr/publications.html

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Kontsevich and D

M. Kontsevich and D. Zagier, Periods, in: Mathematics Unlimited – 2001 and beyond, Springer, 2001, 771–808

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Lepetit, Le th´ eor` eme d’Andr´ e-Chudnovsky-Katz au sens large, North-West

G. Lepetit, Le th´ eor` eme d’Andr´ e-Chudnovsky-Katz au sens large, North-West. Eur. J. Math. 7 (2021), 83–149

work page 2021

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A. B. Shidlovskii, Transcendental numbers , de Gruyter Studies in Math. 12, de Gruyter, Berlin, 1989

work page 1989

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Siegel, ¨Uber einige Anwendungen diophantischer Approximationen, vol

C. Siegel, ¨Uber einige Anwendungen diophantischer Approximationen, vol. 1 S. Ab- handlungen Akad. , Berlin, 1929. St´ ephane Fischler, Universit´ e Paris-Saclay, CNRS, Laboratoirede math´ ematiques d’Orsay, 91405 Orsay, France. Tanguy Rivoal, Institut Fourier, Universit´ e Grenoble Alpes, CNRS,CS 40700, 38058 Greno- ble cedex 9, France. Keywords: E-funct...

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