The workshop explores algebraic and geometric foundations as well as symbolic algorithms for questions related to differential and functional equations, as well as their applications.
Favorite topics have included differential algebra and elimination, Lie symmetry, differential invariants, moving frames, integrability, differential Galois theory, local and closed form solutions, normal form algorithms, operator algebras and combinatorics.
We wish this workshop to be a forum for ideas, techniques and applications . We thus would like to encourage the speakers to contextualize their contribution for a diverse audience.
Other workshop related with differential equations mostly focus on numerical or probabilistic methods; Symbolic Analysis in the main forum for symbolic treatment of differential and functional equations.
The FOCM conference as a whole should provide several other areas of interest for participants of the Symbolic Analysis workshop.
Organizers
JKU Linz
Universidad Politécnica de Madrid
University of Limoges
Speakers
Semi-plenary speakers
University of Illinois Chicago
JKU Linz
Invited speakers
University of Vienna
University of Burgundy
Chinese Academy of Science
University of Illinois Chicago
University of Vienna
Universidad Politécnica de Madrid
INRIA
North Carolina State University
Ben-Gurion University of the Negev
Chinese Academy of Sciences
Institute Polytechnique de Paris
City University of New York
Grenoble Alps University
RICAM Linz
RWTH Aachen
University of Kassel
CNRS
Universidad Autonoma de Madrid
Thursday, 16 July
![[HS11 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/HS11.png){kind=link}
14:00-15:00 semi-plenary talk
Joel Nagloo (University of Illinois Chicago)
In this talk I will give an overview of the work that has been done over the past decade (and maybe even earlier) studying classical algebraic differential equations using various tools from model theory. The central problem we focus on is the classification of all algebraic relations between the solutions of a given equation. We will describe some of the major results and open problems.
15:00-15:30
James Freitag (University of Illinois Chicago)
15:30-16:00
Matthias Aschenbrenner (University of Vienna)
Hardy fields are a convenient setting for “tame” asymptotic analysis. Their formal counterparts are differential fields of transseries. We recently proved a theorem which permits the transfer of logical statements between structures of this kind. I will focus on the special role played by second-order linear differential equations in this story, as well as applications of our main result to such equations. (Joint work with L. van den Dries and J. van der Hoeven.)
16:00-16:30 Coffee Break
16:30-17:00
Evelyne Hubert (INRIA)
Invariants are essential for classifying mathematical objects up to a group of transformations. The action of the orthogonal group in dimension 3 is of particular relevance in applications. The standard action on 3 dimensional space induces actions on tensors, either symmetric or partially so. I will present constructions of invariants for some relevant classes of those. Generating and separating sets of rational invariants are to be characterized by their restrictions to a Seshadri slice. These restrictions are invariant under the 2 dimensional orthogonal group or the octahedral group and explicitly given by binomials or trinomials. The invariants can then be deduced with symbolic computations, but their numerical evaluation can be achieved more robustly using their restrictions. The exhibited set of invariants futhermore allows us to solve the inverse and rewriting problem. The material presented in this talk is based on the results of collaborations with Paul Görlach (Otto von Guericke University Magdeburg) and Martin Jalard (Inria Côte d’Azur).
17:00-17:30
Irina Kogan (North Carolina State University)
I will start with a brief overview of Ovsyannikov’s “podmodeli” (submodels) program (1994) for the symmetry analysis of the gas dynamics system in three spatial dimensions with various choices of the equation of state prescribing pressure as a function of density and entropy. I will then present a contribution to this program jointly with Dilara Siraeva where we considered the case of a specific equation of state prescribing pressure as the sum of entropy and an arbitrary function of density. Such a system has a 12-dimensional symmetry Lie algebra. For a large subset of previously unconsidered, non-similar four-dimensional subalgebras, we computed a complete set of generating invariants. Additionally, we matched each of the subalgebras with its isomorphism class according to the classification of Patera and Winternitz (1977), laying the groundwork for future study of the hierarchy of the reduced systems. For one of the subalgebras, we constructed and explicitly solved a partially symmetry-reduced system (a submodel). This yielded new families of explicit solutions for the original system, whose trajectories we analyzed.
17:30-18:00
Rafael Hernández Heredero (Universidad Politécnica de Madrid)
The talk will begin with an overview of the symmetry-based approach to the study of integrability in systems of evolution equations. We will present recent results, including a new classification of a family of integrable systems, and discuss the computational aspects and challenges of this approach.
18:00-18:30
Joris van der Hoeven (CNRS)
The Hilbert Nullstellensatz provides a bridge between algebra and geometry, which becomes all the more concrete for examples of algebraically closed fields like the complex numbers. What are analogues of this situation in differential algebra?
In our talk, we will survey closed function spaces for the resolution of different types of differential equations: linear partial differential equations, asymptotic differential equations, recursive linear differential equations, and non-linear ordinary differential equations. We will also show some applications of having these concrete examples of closed function spaces.
Friday, 17 July
14:00-14:30
Marina Poulet (Grenoble Alps University)
Mahler equations are functional equations related to many areas such as automata theory. For example, the generating series of automatic sequences are solutions of such equations. When the Mahler equation is regular singular at 0, the solutions can be expressed using only Puiseux series and solutions of equations with constant coefficients. The aim of this talk is to describe an algorithm that determines whether a Mahler equation is regular singular at 0 or not.
This is a joint work with Colin Faverjon.
14:30-15:00
Clemens Raab (RICAM Linz)
Mahler equations are recurrence equations that relate values of functions at places determined by exponentiation. Originally, they were introduced by Kurt Mahler in the context of analytic number theory, but they also play a role in other areas like finite automata. Algorithms to compute rational solutions and series solutions of scalar linear Mahler equations are known. For systems, algorithms for series solutions have been given as well.
In this talk, we present algorithms for computing the rational solutions of linear first-order Mahler systems without uncoupling the system. Refined universal denominators are based on two special left inverses of the Mahler operator on polynomials. For computation of the numerators, we bring the system into a special form and consider transformations to make computations easier. Illustrating examples will be given as well. This is joint work in progress with Moulay A. Barkatou.
15:00-15:30
Ruyong Feng (Chinese Academy of Science)
Let k be an algebraically closed field of characteristic zero, and let B be a finitely generated integral domain over k. We study families of linear differential equations whose coefficients depend algebraically on parameters ranging over an algebraic variety associated with B. Specializing the parameters yields a corresponding family of differential equations.
A natural problem is to understand how algebraic properties of the solutions vary under specialization. In particular, one may ask for which parameter values a specialized equation admits a basis of Liouvillian solutions, even though the original equation does not. The set of such parameter values is called the exceptional set.
In this talk, generalizing a result of Ehud Hrushovski, we show that the exceptional set is suitably “small” in a precise geometric sense. As an application, we prove Matzat’s conjecture in full generality: the absolute differential Galois group of a one-variable function field equipped with a nontrivial derivation is a free proalgebraic group of rank equal to the cardinality of the base field.
This is joint work with Michael Wibmer from University of Leeds.
15:30-16:00
Florian Fürnsinn (University of Vienna)
Deciding whether a solution of a linear differential equation with polynomial coefficients over the rational numbers also satisfies a polynomial relation, i.e., is algebraic, is an old problem going back to the 19th century, to Liouville and Fuchs. In my talk, I will focus on a rather unconventional, arithmetic, approach to this question.
To such a differential equation one can attach, for all prime numbers p, a matrix, called the p-curvature. The Grothendieck p-curvature conjecture asserts that the algebraicity of a full basis of solutions of such a differential equation is equivalent to the vanishing of the p-curvatures for almost all prime numbers p. In 1974 Honda provided a proof of this conjecture for order one equations, by reducing the problem to a theorem of Kronecker, which provides a local-global criterion for the splitting of polynomials over the rational numbers. In 1985 Chudnovsky and Chudnovsky gave a new proof of Kronecker’s result, and with it of Honda’s result, using Padé approximation.
I will explain how to use the proof of the Chudnovsky brothers to make Honda’s result effective. More precisely, given a linear differential equation of order one with polynomial coefficients over the rational numbers I will deduce an upper bound on the number of p-curvatures to be computed in order to decide the algebraicity of all solutions of the equation.
This talk is based on joint work with Lucas Pannier.
16:00-16:30 Coffee Break
16:30-17:00
Werner Seiler (University of Kassel)
Using the Thomas-Fermi and the Poisson-Boltzmann equations as test cases, we study the use of symmetries in the analysis of singularities of ordinary differential equations.
17:00-17:30
Wei Li (Chinese Academy of Sciences)
We analyze the behavior of systems of algebraic differential equations when considered as systems of difference-differential equations, with special emphasis on systems which define strongly minimal sets relative to the theory DCF_{0,m} of differentially closed fields of characteristic zero with m distinguished commuting derivations. We show that if X is a strongly minimal set relative to DCF_{0,m} defined by a finite system of algebraic partial differential equations and the forking geometry on X is trivial, then X remains minimal when regarded as definable set relative to the theory DCFA_{0,m} of difference-differentially closed fields of characteristic zero with m commuting derivations. In particular, when X is totally disintegrated or strictly minimal, the possible difference-differential equations or purely difference equations compatible with X is extremely restricted. We illustrate these results with several computational examples. This talk reports on joint work with Thomas Scanlon.
17:30-18:30 semi-plenary talk
Veronika Pillwein (JKU Linz)
It is well known that holonomic expressions can be represented by a finite amount of data, namely linear difference or differential equations with polynomial coefficients, together with initial values. Furthermore, a multitude of algorithms has been developed and implemented to facilitate calculations involving these objects. Recently, we have been working on extensions of some of these algorithms to objects satisfying linear difference or differential equations with holonomic coefficients. In this talk, we give an overview of the current status of this research.
Saturday, 18 July
14:00-14:30
Yulia Mukhina (Institute Polytechnique de Paris)
Differential parametric dynamical systems appear in many applications in modeling and control theory. Frequently, the task is to compute the minimal differential equation satisfied by experimentally observed data, which is an important special case of the more general differential elimination problem.
In this talk, we present how to tackle this computation in an evaluation-interpolation framework, gaining efficiency over classical methods by avoiding intermediate expression swell.
First, we provide explicit bounds on the Newton polytope of the minimal equation in terms of the degrees of the dynamical system, and prove these bounds to be sharp in more than half the cases. Second, we show how convex geometry can be further exploited in the sparse setting.
Finally, we show how to use the structure of input dynamical systems by designing an algorithm that, through a careful choice of evaluation points, reduces the problem to solving a block-triangular linear system.
Based on joint work with Gleb Pogudin, Rafael Mohr and Maxime Despréaux.
14:30-15:00
Daniel Robertz (RWTH Aachen)
Let $G$ be a classical group of dimension $d$ and let $\boldsymbol{a} = (a_1, …, a_d)$ be differential indeterminates over a differential field $F$ of characteristic zero with algebraically closed field of constants $C$. Further let $A(\boldsymbol{a})$ be a generic element in the Lie algebra $\mathfrak{g}(F\langle \boldsymbol{a} \rangle)$ of $G$ obtained from parametrizing a basis of $\mathfrak{g}$ with the indeterminates $\boldsymbol{a}$. It is known that the differential Galois group of $\boldsymbol{y}’ = A(\boldsymbol{a}) \boldsymbol{y}$ over $F\langle \boldsymbol{a} \rangle$ is $G(C)$. In this talk we report on recent joint work with Matthias Seiss in which we constructed a differential field extension $\mathcal{L}$ of $F\langle \boldsymbol{a} \rangle$ such that the field of constants of $\mathcal{L}$ is $C$, the differential Galois group of $\boldsymbol{y}’ = A(\boldsymbol{a}) \boldsymbol{y}$ over $\mathcal{L}$ is still the full group $G(C)$ and $A(\boldsymbol{a})$ is gauge equivalent over $\mathcal{L}$ to a matrix in normal form introduced by Seiss. We also consider specializations of the coefficients of $A(\boldsymbol{a})$ as well as determining the differential Galois group of the specialized equation.
15:00-15:30
Alexey Ovchinnikov (City University of New York)
Parameter estimation is a key problem in analyzing parametric ODE models. These estimations are based on measuring output data of the model. However, measurements themselves can be prohibitively expensive, creating a need to minimize the number of measurements while maintaining high estimation accuracy. It is a challenging problem to determine best measurement time points based on the user’s constraints. In the talk, we will discuss how this problem can be approached.
15:30-16:00
Marian Zurro (Universidad Autonoma de Madrid)
This talk addresses the Galoisian study of integrability for nonlinear partial differential equations, a problem that remains largely open within the framework of differential Galois theory. Our approach is motivated by the classical example of the Korteweg–de Vries (KdV) equation [3], whose integrability is understood through its associated Lax pair — a pair of differential operators for which the KdV equation arises as the compatibility condition. A foundational contribution in a Galoisian study was made by Morales-Ruiz, Rueda, and Zurro [1], who developed a spectral Galois theory for the Schrödinger operator with stationary KdV potential. Building on their work, and on the algebro-geometric study of third-order operators carried out by Rueda and Zurro [2], we present a Galoisian approach to the variational equation associated with a KdV-type evolution equation in 1+1 dimensions, exploiting the spectral techniques arising from third-order operator theory. As an application, we perform explicit computations of closed-form solutions to the variational equation of KdV linearized around a cnoidal wave, highlighting the role of Galoisian methods in their derivation.
I will present recent results and open questions related with this problems, in the framework of the project Algorithmic Differential Algebra and Integrability (ADAI).
This work is part of an ongoing collaboration with J. J. Morales-Ruiz (Universidad Politécnica de
Madrid, Spain) and J. P. Ramis (Université Toulouse III, France).
References
[1] J. J. Morales-Ruiz, S. Rueda, and M. A. Zurro, Spectral Picard–Vessiot fields for Algebro-geometric Schrödinger operators, Ann. Inst. Fourier, 71 (2021), no. 3, 1287-1324.
[2] S. Rueda and M. A. Zurro, Spectral Curves for Third-Order ODOs, Axioms, 13, (2024), no. 4, 274.
[3] D. J. Korteweg and G. de Vries, On the change of form of long waves advancing in a rectangular canal, Philos. Mag., 39 (1895), 422–443
16:00-16:30 Coffee Break
16:30-17:00
Thierry Combot (University of Burgundy)
Consider a derivation $D=A\partial_x+ B \partial_y$ with A,B polynomials. A Darboux polynomial is a polynomial P such that P divides D(P), and corresponds to an algebraic trajectory. We present an algorithm, which given a degree bound N, computes the irreducible Darboux polynomials of degree up to N in time O(N^(1+w)). The proof also gives an improvement of Jouanolou Theorem on the number of Darboux polynomials required to build a first integral.
17:00-17:30
Partha Kumbhakar (Ben-Gurion University of the Negev)
A solution of an algebraic differential equation
f(y, y′, . . . , y(n)) = 0,
where f is a polynomial in n+1 variables with coefficients in the differential field (C(x), d/dx), is called classical if it can be obtained from C(x) by repeatedly adjoining algebraic functions, solutions of linear differential equations, and abelian functions (for example, elliptic functions).
Inspired by the work of Umemura and Nishioka, we study the following question: When does an algebraic differential equation admit a classical solution? We answer this question by characterizing the differential field structure of the associated solution field, namely, the differential
field generated over C(x) by a solution of the equation.
Our approach is based on differential Galois theory, in particular the theory of strongly normal extensions introduced by Kolchin. We obtain a characterization of arbitrary differential subfields of strongly normal extensions, which forms the main ingredient in our analysis of solution fields.
As an application, we generalize a theorem of Goldman and Singer, which characterizes first-order algebraic differential equations admitting solutions of linear differential equations, to higher-order algebraic differential equations. Finally, we relate our results to the modeltheoretic notions of internality and almost internality to the constants, and answer a question of Jin and Moosa.
17:30-18:00
Jacques-Arthur Weil (University of Limoges)
Posters
![[PZ11 – 2nd Floor]](https://focm2026.univie.ac.at/wp-content/uploads/2026/06/poster_zones_2-stock.png){kind=link}
Partha Kumbhakar
(Ben-Gurion University of the Negev)
An autonomous first-order algebraic differential equation
f(y, y′) = 0,
over an algebraically closed field of constants C can be naturally associated with a pair (X, ω),
where X is a smooth projective curve and ω is a meromorphic 1-form on X. This correspondence
allows one to study autonomous first-order algebraic differential equations through the geometry
of pairs (X, ω). The pairs (X, ω) are classified into four types: exact, exponential, Weierstrass,
and general type. A pair (X, ω) is said to be new if it is not obtained as the pullback of another
pair under a nontrivial morphism of curves; otherwise, it is said to be old.
Equations of new and general type are distinguished by remarkable algebraic properties.
First, their solutions cannot be expressed in terms of algebraic functions, solutions of linear
differential equations, abelian functions, or solutions of other nonlinear first-order differential
equations. Consequently, they give rise to ”new” special functions. Second, any set of distinct
nonconstant solutions of new and general type equations is C-algebraically independent.
Using techniques from the theory of algebraic curves, we study the existence of new and
general type meromorphic 1-forms on curves through explicit constructions. Specifically, we
construct large families of new and general type meromorphic 1-forms on P1, elliptic, and
hyperelliptic curves. We also establish a connection with the Hurwitz realization problem for
branched covers of the Riemann sphere, which leads to an algorithm for determining whether
differential equations of the form y′ = f(y), where f or 1/f is a Laurent polynomial, are new
or old.
Orla McGrath
(University of Leeds)
While an (affine) algebraic group is a subgroup of a general linear group defined by some polynomials, an (affine) difference algebraic group is a subgroup of a general linear group defined by some difference polynomials. We develop the theory of difference algebraic groups in the case where we have finitely many pairwise commuting difference operators. We show that any difference algebraic group can be defined as a difference closed subgroup of the general linear group by finitely many difference polynomials, and this result allows us to prove the existence of a dimension polynomial for any partial difference algebraic group.
Mattia Puddu
(RWTH Aachen)
We describe a criterion and implement the corresponding algorithm in OSCAR to establish whether a difference polynomial belongs to certain difference ideals associated with a quasi-simple system S of polynomial difference equations f_1 = … = f_s = 0 and inequations q_1
eq 0, …, q_t
eq 0 in the difference polynomial ring k{y_1, …, y_n} with a fixed ranking <.
A quasi-simple system requires that the set F = {f_1, …, f_s} be triangular and passive with respect to <.
This ensures that a normal form algorithm is well-defined: for any difference polynomial p, there exist difference polynomials q and r such that qp-r lies in the difference ideal E generated by f_1, …, f_s; r is called the normal form of p with respect to F and <.
Starting from S, we investigate systems to which we add shifts of the original equations and inequations.
For a fixed variable v, and assuming that the ranking is orderly, there are only finitely many variables <=v.
It follows that a system of difference polynomial equations and inequations whose defining polynomials have leaders <=v is finite, and we may regard it as an algebraic system in a polynomial ring in finitely many variables.
The study of these systems is closely related to the study of the difference variety of the difference ideal E:Q, where Q denotes the multiplicatively closed set generated by all shifts of the inequations in S.
The algorithm determines whether a polynomial p lies in this difference ideal in terms of its normal form.
Chitrarekha Sahu
(Indian Institute Of Science Education And Research Mohali)
We will present a generalization of Michael Singer’s results (Singer, 1992) to higher-dimensional vector fields. While a recent generalization by Aziz et al. (2021) relies on Puiseux series and differential forms, our approach utilizes Differential Galois Theory to prove that if a Liouvillian first integral exists, there must be one in a simplest form that also satisfies a linear differential equation. We present several examples of systems having first integrals in Picard-Vessiot extensions that are not Liouvillian.
We further establish a necessary condition for the existence of an elementary first integral for linear differential equations with unipotent differential Galois groups. Additionally, we extend Singer’s theorem (1977) for iterated integrals to the broader domain of generalized iterated integrals.
The results presented here are based on ongoing collaborative projects with Partha Kumbhakar and Varadharaj R. Srinivasan.
Werner M. Seiler
(University of Kassel)
Conservation laws play an important role in both the theory and applications of dynamical systems. In the literature, several approaches to discovering conservation laws of finite-dimensional dynamical systems using neural networks as basic technology have been reported. For obtaining a closed-form expression for a discovered conservation law, a subsequent symbolic regression step is necessary. One problem here is that both the training of the neural network and the symbolic regression are computationally very expensive and large amounts of data are necessary for the training. We describe another approach to machine learning conservation laws from trajectory data using kernel methods, more precisely an indeterminate form of kernel ridge regression. Using an inhomogeneous polynomial kernel, we obtain directly polynomial conservation laws in an explicit closed-form without a subsequent symbolic regression. Our approach is not only computationally much more efficient than neural networks (cubic complexity in the number of data points), but also requires much less training data (often a few dozens to a few hundred data points suffice). This point becomes important, if one is working with experimental data, which may be much harder to get in large quantities. We also show how one can discover several, functionally independent conservation laws from a single data set and we present a method for making the obtained conservation laws sparse which significantly improves the interpretability of the results.
This is joint work with the two Ph.D. students Meskerem Abebaw Mebratie and Rüdiger Nather and with Guido von Rudorff.
This poster presents an algorithmic approach to the structural identifiability and observability (SIO) analysis of control systems governed by nonlinear differential equations based on symmetry analysis. In the mathematical modeling of dynamic systems, a fundamental question is whether the unknown parameters of a model can be uniquely determined from input-output data. This property is known as structural identifiability. Closely related is observability, which asks whether the internal state variables can be reconstructed from the same data. While continuous (Lie) symmetries are well known to obstruct local SIO, the role of discrete symmetries remains unexplored. In this work, we demonstrate that discrete symmetries are the origin of parameters that are structurally locally identifiable but not globally (SLING).
To address this, we work with the finite determining system associated with such symmetries, which is generally a highly nonlinear system. We use rigorous techniques from differential algebra, in particular the Thomas decomposition, which allows for a decomposition of the solution space into simple systems. In order to decide SIO, it is not necessary to explicitly solve this system but rather detect the existence of nontrivial solutions. This can be done algorithmically with methods from the formal theory of differential equations. The results illustrate how symbolic methods can detect and characterize symmetry-induced ambiguities of state variables and parameters in control systems, providing a tool for the structural analysis of differential equations.
Joint work with Xabier Rey Barreiro, Nick Baberuxki, Mestkerem Abebaw Mebratie, and Alejandro F. Villaverde.