SCDDE 2026: Symbolic Computation and Differential and Difference Equations

Speakers

  • Johannes Blümlein (DESY, Deutsches Elektronen-Sychrotron, Germany)

    Mathematical methods in perturbative quantum field theory and the analytic integration of Feynman integrals
    Abstract: We present recent advances in the analytic calculation of single scale Feynman integrals in renormalizable quantum field theories. We consider complete physical processes at large-scale high energy colliders, such as the anomalous dimensions, massless and massive Wilson coefficients and the massive form factors to three-loop order in QCD. The fundamental methods to be used are difference and differential equations, which have first to be established by guessing methods. The corresponding equations are huge. In the case of first order factorizing equations iterative-integral solutions are obtained. We also present computer-algebraic methods to solve non first order factorizing differential equations.
    Slides

  • Alin Bostan (Inria Saclay and Sorbonne Universite, France)

    Algorithmic determination of algebraic values of E-functions
    Abstract: E-functions form a large class of transcendental entire functions, introduced by Siegel in 1929, which generalize the exponential function. They are extensively studied in number theory because their values enjoy important transcendence results, thanks to a theory developed over decades by mathematicians such as Siegel, Shidlovskii, Nesterenko, Andre and Beukers. One of the culminations of this theory is a very general version of the Hermite-Lindemann-Weierstrass theorem for E-functions. In this talk, the primary focus will be on effective versions of these results. I will present a recent algorithm that, for any given E-function, computes the set of all algebraic numbers to which it takes algebraic values. The algorithm crucially relies on a task called "minimization of linear differential equations", and is efficient in practice.
    Joint work with Tanguy Rivoal and Bruno Salvy.

  • François Boulier (University of Lille, France)

    The DenefLipshitz Algorithm
    Abstract: In their celebrated 1984 article, Denef and Lipshitz prove a theorem which states the existence of an algorithm for deciding whether a finite system of ordinary differential polynomials has formal power series solutions. In a recent joint work with Lemaire and Vu, the algorithmic ideas of Denef and Lipshitz were turned for the first time into a practical algorithm, called DenefLipshitz, implemented in the Python DifferentialAlgebra package, which actually solves this problem for systems of ordinary differential polynomial with algebraic conditions on the series coefficients. This talk aims at presenting this algorithm.
    Paper
    DifferentialAlgebra package

  • Shaoshi Chen (AMSS, Chinese Academy of Sciences, China)

    Symbolic Integration of Differential Forms: from Abel to Zeilberger
    Abstract: This talk focuses on symbolic integration of differential forms, with a particular emphasis on historical and modern developments, from Abel\u2019s addition theorems for Abelian integrals to Zeilberger\u2019s creative telescoping for parameterized integrals. It explores closed rational p-forms and provides algorithmic approaches for their integration, extending classical results like Hermite reduction and Liouville\u2019s theorem. The integration of closed differential forms with parameters is further examined through telescopers, offering a unified framework for handling both algebraic and transcendental cases. This talk is based on a joint work with David A. Cox and Yisen Wang.

  • Thomas Cluzeau (University of Limoges, France)

    Polynomial solutions for general linear polynomial ordinary integro-differential systems and applications

    Abstract: In this talk, I will first review the main properties of the ring of ordinary integro-differential operators with polynomial coefficients. Next, I will address the problem of computing polynomial solutions to general linear systems of ordinary integro-differential equations with polynomial coefficients. I will also briefly explain that this algorithmic problem is a key step for many computations involving matrices whose elements are linear integro-differential operators, such as the computation of left/right syzygies, left/right inverses, left/right factorizations, and thus for the development of an efficient algebraic analysis approach to linear systems of ordinary integro-differential equations using efficient elimination methods and efficient homological algebra. The linear systems that appear in the above problems are generally rectangular and non-homogeneous. I will describe the first algorithm for computing polynomial solutions of non-homogeneous rectangular systems of linear integral-differential equations with polynomial coefficients. The algorithm is implemented in the freely available Maple package Bavula, This work is based on a collaboration with Alban Quadrat (Inria Paris, Ouragan, Sorbonne Univ., France) as part of the PhD thesis of our former PhD student Camille Pinto.

  • Manuel Kauers (Johannes Kepler University Linz, Austria)

    A proof of Conjecture 16
    Abstract: A few years ago, Koutschan and the speaker proposed a number of conjectures concerning the D-Finiteness of certain entries of the OEIS. Since then, several of these conjectures have been proven. In the talk, we present one such proof, obtained in joint work with Frederic Chyzak and Hui Huang. Our proof settles Conjecture 16, stating that the counting sequence of (1,3)-regular graphs where the number of vertices exceeds the number of edges by 1 is D-finite. Our proof relies on the theory of symmetric D-finite functions, pioneered by Gessel, as well as (of course) the technique of creative telescoping, pioneered by Zeilberger.

  • Markus Lange-Hegemann (Technische Hochschule Ostwestfalen-Lippe, Germany)

    Differential Algebraic Machine Learning in Linear PDE Solution Spaces
    Abstract: We consider linear constant-coefficient PDE systems. We present a computational differential-algebraic approach to probabilistic and learning-based models that work directly within their solution spaces. The starting point is the Ehrenpreis–Palamodov fundamental principle, which describes solutions through algebraically computable characteristic varieties and exponential-polynomial Fourier-type modes. We use this representation to construct two learning approaches within the solution spaces of such PDEs. First, it yields Ehrenpreis–Palamodov Gaussian processes: Bayesian priors concentrated on admissible PDE solutions, supporting conditioning on sparse, noisy, initial, or boundary data. Second, it yields exact-by-construction trainable shallow neural networks whose hidden units are themselves solutions of the governing equations, such as Maxwell’s equations or the wave equation. The main message is that symbolic structure can make scientific machine learning both more faithful and more efficient: rather than penalizing violations of PDEs after the fact, we learn directly in a computationally tractable space of exact solutions. We illustrate these ideas in applications to PDE solving, system control, and sampling from solution spaces.

  • Stefan Müller (University of Vienna, Austria)

    From reaction networks to "positive algebraic geometry" - and back
    Virtually every power-law and polynomial dynamical system used across chemistry, biology, engineering, and economics can be modeled as a reaction network governed by generalized mass-action kinetics. Over the past decade, we focused on complex-balanced equilibria, which are uniquely determined by the network structure. As one key result, we characterized the unique existence of these equilibria using sign vectors of subspaces derived from stoichiometric coefficients and kinetic orders. Moving beyond complex balance, we explore general positive equilibria by analyzing parametrized systems of generalized polynomial equations. We abstract the problem to identify its core geometric components, specifically the coefficient polytope and the monomial dependency subspace. Our main result demonstrates how to rewrite these polynomial equations as binomial equations on the coefficient polytope, and how the monomial dependency determines the problem complexity. As intended, this geometric framework yields immediate applications in real fewnomial theory and kinetic systems.
    Joint work with Georg Regensburger.

  • Peter Paule (RISC, Johannes Kepler University Linz, Austria)

    The Unreasonable Effectiveness of Computer Algebra in the Mathematical Sciences
    Abstract: Despite the current renaissance of AI, the main theme of the talk is on more traditional lines: namely, to stress the huge potential of algorithmic mathematics, and of respective computer algebra software, for applications in pure mathematics and related fields. For example, the Ramanujan Machine (Nature 590, 2021) creates mathematical conjectures using AI and computer automation. On the other hand, Cristian-Silviu Radu (RISC) developed a computer algebra algorithm which can be used to discover (and prove!) identities even Ramanujan would have appreciated to see. In the talk we present a variety of such examples from different areas: optimization of antenna radiation, special functions and Gauss' contiguous relations, linear Diophantine inequalitites and partitions of numbers, symbolic summation in quantum field theory, a.s.o.

  • Clemens Raab (RICAM, Austrian Academy of Sciences, Linz)

    Integro-differential rings and operators with evaluation at singularities
    Roughly 20 years ago, integro-differential algebras and integro-differential operators were introduced in Symbolic Computation by Rosenkranz and Regensburger to treat boundary value problems. Originally designed for smooth functions, point evaluations play a key role and the potential to deal with singularities is limited. In this talk, we present joint work with Georg Regensburger that generalizes these algebraic structures to treat functions with singularities. This is based on a generalized notion of evaluation, which gives rise to additional terms in well-known identities like the Taylor-formula. Integro-differential operators with generalized evaluation have normal forms that are more involved. Furthermore, we analyzed the algebraic structure behind nested integrals independent of the type of integrands or their singularities. This allows to construct the integro-differential ring generated by a commutative differential ring and to compute with normal forms of nested integrals. Such nested integrals arise when solving reducible differential equations, for example.