# Meeting of the project De rerum natura

The meeting will happen virtually.

2021/06/03–04: general meeting (Strasbourg). The meeting (Thu afternoon–Fri morning) happens in the continuation of the 2021 meeting of GDR EFI at the beginning of the week (Mon–Wed). We will have three speakers, at https://webconf.math.cnrs.fr/b/dre-te9-3zj:

• Thursday 3, 14:00, Alexandre Goyer, Symbolic-numeric factorization of linear differential operators, slides.
I will talk about the factorization of differential operators of the type $L = a_n\partial^n + \dots + a_1\partial + a_0$ with $a_i \in {\mathbb Q}(z)$. More specifically, I will follow an approach proposed by van der Hoeven in 2007. Under a regularity assumption, van der Hoeven suggests to start with the monodromy group of the operator to find a subspace invariant under the action of the differential Galois group. The originality of his method comes from contribution of numerical computation in a symbolic context. I will outline this algorithm and discuss its implementation. The subtasks of the computation of approximate invariant subspaces and the guessing of rational fractions with Padé-Hermite approximants will be addressed. Joint work with Frédéric Chyzak and Marc Mezzarobba.
• Friday 4, 10:00, Gwladys Fernandes, A Galoisian proof of Ritt theorem on the differential transcendence of Poincaré functions, slides.
In 1890, J. F. Ritt gave the complete list of all the algebraically differential solutions of the Schröder's equation: \begin{equation*} R(y(t))=y(qt), \end{equation*} where $R(t)\in{\mathbb C}(t), R(0)=0$, $R'(0)=q\in{\mathbb C}$, $|q|>1$. The proof of the author is quite technical. I will present in this talk a recent work in collaboration with L. Di Vizio, in which we establish a new proof of Ritt's result, based on Galois theory for functional equations. Ritt's result is central in further classifications of differentially algebraic solutions of other types of difference equations, and shares links with dynamical systems.
• Friday 4, 11:15, Sandro Franceschi, Algebraic nature of the stationary distribution of a reflected Brownian motion in a cone, and Tutte's invariants, slides.
We consider the classical problem of determining the stationary distribution of reflected Brownian motion in a two dimensional wedge. We study the algebraic and differential nature of the Laplace transform of this stationary distribution. We derive necessary and sufficient conditions for this Laplace transform to be rational, algebraic, differentially finite or more generally differentially algebraic. In the differentially algebraic case, we go further and provide an explicit, integral-free expression.
To prove these results, we start from a functional equation that the Laplace transform satisfies, to which we apply tools from diverse horizons. To establish differential algebraicity, a key ingredient is Tutte’s invariant approach, which originates in enumerative combinatorics. It allows us to express the Laplace transform as a rational function of a certain canonical invariant, a hypergeometric function in our context. To establish differential transcendence, we turn the functional equation into a $q$-difference equation and apply Galoisian results on the nature of the solutions to such equations.
Joint work with Mireille Bousquet-Mélou, Andrew Elvey Price, Charlotte Hardouin, and Kilian Raschel.