### Workshop B3 - Symbolic Analysis

Organizers: Bruno Salvy (INRIA & École Normale Supérieure de Lyon, France) - Jacques-Arthur Weil (Université de Limoges, France) - Irina Kogan (North Carolina State University, USA)

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## Talks

July 13, 14:30 ~ 14:55 - Room B2

## Invariants of ternary quartics under the orthogonal group

### Inria Méditerranée, France   -   evelyne.hubert@inria.fr

Classical invariant theory has essentially addressed the action of the general linear group on homogeneous polynomials. Yet the orthogonal group arises in applications as the relevant group of transformations, especially in 3 dimensional space. Having a complete set of invariants for its action on quartics is, for instance, relevant in determining biomarkers for white matter from diffusion MRI [1]

We characterize a generating set of rational invariants of the orthogonal group by reducing the problem to the action of a finite subgroup on a slice. The invariants of the orthogonal group can then be obtained in an explicit way. But their numerical evaluation can be achieved more straightforwardly. These results can furthermore be generalised to sextics and higher even degree forms thanks to a novel and explicit construction of a basis of harmonic polynomials at each degree.

[1] A.~Ghosh, T.~Papadopoulo, and R.~Deriche. Biomarkers for hardi: 2nd \& 4th order tensor invariants. In {\em {IEEE International Symposium on Biomedical Imaging: From Nano to Macro - 2012}}, 2012.

Joint work with Paul Görlach (Inria \& Max Planck Institute Leipzig), Théo Papadopoulo (Inria).

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July 13, 15:00 ~ 15:25 - Room B2

## Formal recursion operators of integrable PDEs of the form $q_{tt}=F(q,q_x,q_t,\ldots)$

### Rafael Hernández Heredero

We will explain how the symmetry approach to integrability applies to partial differential equations of the form $q_{tt}=F(q,q_{x},\ldots,q_{n},q_{t},q_{tx},\ldots,q_{tm}).$ Any such equation is integrable if it admits a formal recursion operator, i.e. a pseudodifferential operator $\mathcal{R}$ of the form $\mathcal{R}:= L+MD_{t}$ where $L$ and $M$ are pseudodifferential operators in the derivation $D_x$ satisfying the symmetry condition $\mathcal{F}(L+MD_{t})=(\overline{L}+MD_{t})\mathcal{F}.$ Here $\overline{L}:= L+2M_{t}+[M,V]$ and $\mathcal{F}=D_t^2-U-VD_t$ is the linearization operator of the PDE, i.e. the operator appearing in its variational equation, so $U=u_{n}D^{n}+u_{n-1}D^{n-1}+\cdots+u_{0}$ $V=v_{m}D^{m}+v_{m-1}D^{m-1}+\cdots+v_{0}$ with $u_{i}=\frac{\partial F}{\partial q_{i}},\quad v_{j}=\frac{\partial F}{\partial q_{tj}}.$

We are then confronted with solving an equation over pseudodifferential operators in two derivations, a rather nontrivial problem. The equation happens to have a somewhat triangular structure, making its resolution possible. But in the solving process there appear obstructions, written as conditions over the rhs $F$ of the PDE, that are interpreted as integrability conditions.

The algebra of formal recursion operators has an interesting structure, and it has important relationships to algebras of commuting (pseudo)-differential operators in two derivations.

Some previous results and background were presented in Rafael Hernández Heredero, V. V. Sokolov and A. B. Shabat, A new class of linearizable equations, J. Phys. A: Math. Gen. 36 (2003) L605–L614".

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July 13, 15:30 ~ 15:55 - Room B2

## Computing the Galois-Lie Algebra of Completely Reducible Differential Systems.

### Université de Limoges, France   -   weil@unilim.fr

This talk is related to the work presented by Thomas Dreyfus earlier in this session.\\ A linear differential system $[A]: Y'=AY$ is called completely reducible when $A$ is block-diagonal and each block is the matrix of an irreducible system. We show how to compute the Lie algebra $\mathfrak{g}$ of the Galois group of $[A]$. \\ We will first explain how to represent a copy of $\mathfrak{g}$ as a subsystem of a system constructed on $[A]$. We show how to guess this subsystem by using modular techniques or invariants. Validation of this guess is achieved through, first, computing an explicit conjugacy between Lie algebras and finally finding algebraic solutions of a (small) linear differential system.

Joint work with Moulay Barkatou, Thomas Cluzeau (Université de Limoges, France) and Lucia Di Vizio (Université de Versailles, France).

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July 13, 16:00 ~ 16:25 - Room B2

## Singular initial value problems for quasi-linear ordinary differential equations

### Universität Kassel, Germany   -   seiler@mathematik.uni-kassel.de

We discuss the existence, (non-)uniqueness and regularity of solutions of initial value problems for quasi-linear ordinary differential equations where the initial data corresponds to a singularity of the equation. We show how the Vessiot distribution permits to reduce the question of existence and (non-)uniqueness to the study of the local solution behaviour of a vector field near a stationary point. For regularity results a similar analysis must be performed for different prolongations of the given equation.

Joint work with Matthias Seiß (Universität Kassel, Germany).

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July 13, 17:00 ~ 17:25 - Room B2

## Multispace and variational methods

### University of Kent, UK   -   E.L.Mansfield@kent.ac.uk

Recently with Gloria Mari Beffa, a multi-dimensional version of multispace was constructed. This incorporates Lagrange and Hermite interpolations of functions but has the standard jet bundle embedded as a smooth submanifold. The construction allows for functional approximations and their smooth continuum limits to be studied simultaneously. Applications include invariants of Lie group actions via a moving frame on multispace, smooth and discrete integrable systems, and smooth and discrete variational calculus including Noether's theorem. Further, there is a multispace version of the preserved symplectic form for a variational system, allowing smooth and discrete Hamiltonian versions of Euler Lagrange equations to be studied simultaneously. The talk will include joint work with Gloria Mari Beffa and Peter Hydon, and perhaps others, depending on the particular application of the multispace variational calculus developed by the time of the conference.

Joint work with Gloria Mari Beffa (UW, Madison) and Peter Hydon (University of Kent).

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July 13, 17:30 ~ 17:55 - Room B2

## Non-integrability of the Armburster-Guckenheimer-Kim quartic Hamiltonian through Morales-Ramis theory

### Universidad Simón Bolívar, Colombia   -   primitivo.acosta@unisimonbolivar.edu.co

In this talk it will be presented the non-integrability of the three-parameter Armburster-Guckenheimer-Kim quartic Hamiltonian using Morales-Ramis theory, with the exception of the three already known integrable cases. Poincaré sections to illustrate the breakdown of regular motion for some parameter values. In particular we obtain conditions to get meromorphic non-integrability results through the Galoisian analysis of the variational equation of the AGK Hamiltonian which is a Legendre differential equation. On the other hand, the complete conditions for rational non-integrability results are presented through the Bostan-Combot-Safey El Din algorithm which is developed in maple.

Joint work with Martha Álvarez-Ramírez (Universidad Autónoma Metropolitana Iztapalapa, Mexico) and Teresa Stuchi (Universidade Federal do Rio de Janeiro, Brazil).

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July 13, 18:00 ~ 18:25 - Room B2

## Computing the differential Galois group using reduced forms

### University Lyon 1, France   -   thomas.dreyfus@ens-cachan.org

To a linear differential system one may associate a group, the differential Galois group, that measures the algebraic relations between the solutions of the system. The latter may be seen as an algebraic group and its computation is in general a difficult task. In this talk we explain how to transform the system so that the new system belong to the Lie algebra of the differential Galois group (such transformation always exists and the new system will be said to be on the Kolchin-Kovacic reduced form). Furthermore, as we will see, the latter reduction will be helpful for computing the differential Galois group.

Joint work with Jacques-Arthur Weil (University of Limoges, France).

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July 14, 14:30 ~ 14:55 - Room B2

## Finite automata, automatic sets, and difference equations

### North Carolina State University, USA   -   singer@ncsu.edu

A finite automaton is one of the simplest models of computation. Initially introduced by McCulloch and Pitts to model neural networks, they have been used to aid in software design as well as to characterize certain formal languages and number-theoretic properties of integers. A set of integers is said to be m-automatic if there is a finite automaton that decides if an integer is in this set given its base-m representation. For example powers of 2 are 2-automatic but not 3-automatic. This latter result follows from a theorem of Cobham describing which sets of integers are m- and n-automatic for sufficiently distinct m and n. In recent work with Reinhard Schaefke, we gave a new proof of this result based on analytic results concerning normal forms of systems of difference equations. In this talk, I will describe this circle of ideas.

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July 14, 15:00 ~ 15:25 - Room B2

## Factorization of stationary Schrodinger operators over KdV spectral curves

### Sonia Rueda

In 1928, J. L. Burchall and T. W. Chaundy established a correspondence between commuting differential operators and algebraic curves. With the discovery of solitons and the integrability of the Korteweg de Vries (KdV) equation, using the inverse spectral methods, their theory found applications to the study of partial differential equations called integrable (or with solitonic type solutions: Sine-Gordon, non linear Schrödinger, etc). Burchall and Chaundy had discovered the spectral curve, which was later computed by E. Previato, using differential resultants. The spectral curve allows an algebraic approach to handling the inverse spectral problem for the finite-gap operators, with the spectral data being encoded in the spectral curve and an associated line bundle.

In this work, we explore the benefits of using differential resultants to compute the Burchall and Chaundy polynomials. We review the definition of the differential resultant of two ordinary differential operators and its main properties. We revisit Enma Previato's result about the computation of the spectral curve of two commuting differential operators using differential resultants. We use these results to establish the appropriate fields were commuting operators have a common factor, which can be computed using differential subresultants.

These results will allow us to give new explanations to some well known results related with the celebrated KdV hierarchy. We will describe the centralizer of a Schrödinger operator $L=-\partial^2+u$, for stationary potentials $u$ subject to constrains given by the KdV hierarchy, using results of Goodearl. We present an algorithm to factor Schrödinger operators $L_s-\lambda=-\partial^2+u_s-\lambda$ with $u_s$ satisfying the KdV${}_s$ equation. Previous results, for hyperelliptic curves, construct factorizations as formulas using $\theta$-functions. The method we are presenting is effective and it points out the fact that closed formulas for factors of the Schrödinger operator over the curve can be obtained using: a global parametrization of the spectral curve (if it exists), and the subresultant formula operator. Furthermore, one can extend the coefficient field of $L_s$ to the field of a spectral curve and then to the Liouvillian extension given by $\Psi'=\phi_s\Psi$, with $\phi_s$ satisfying the Ricatti equation $\phi'+\phi^2=u_s-\lambda$. In this manner, we will describe the Picard-Vessiot fields of $L_s-\lambda$. A spezialization process in $\lambda$ can be adapted to our methods.

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July 14, 15:30 ~ 15:55 - Room B2

## Desingularization of First Order Linear Difference Systems with Rational  Function Coefficients

### XLIM, University of Limoges ; CNRS, France   -   moulay.barkatou@unilim.fr

It is well known that for a first order system of linear difference equations  with rational function coefficients, a solution that is holomorphic in some  left half plane can be analytically continued to a meromorphic solution in the  whole complex plane. The poles stem from the singularities of the rational  function coefficients of the system. Just as for systems of differential  equations, not all of these singularities necessarily lead to poles in a  solution, as they might be what is called removable. In this talk, we show how  to detect and remove these singularities and further study the connection  between poles of solutions, removable singularities and the extension of  numerical sequences at these points. This is a joint work  with Maximilian Jaroschek

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July 14, 16:00 ~ 16:25 - Room B2

## Symbolic computation for preserving conservation laws

### University of Kent, UK   -   G.Frasca-Caccia@kent.ac.uk

The main goal of the field of investigation known as Geometric Integration is to reproduce, in the discrete setting, a number of geometric properties shared by the original continuous problem. Conservation laws are among a PDE’s most fundamental geometric properties. In this talk a new strategy, which uses symbolic computation and is based on the fact that conservation laws are in the kernel of the Euler operator, is used to develop new methods that preserve multiple discrete conservation laws. The new schemes are numerically compared with some other schemes existing in literature.

Joint work with Peter Hydon (University of Kent, UK).

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July 14, 17:00 ~ 17:50 - Room B2

## The linear Mahler equation: linear and algebraic relations

### CNRS and Université de Lyon , France   -   Boris.Adamczewski@math.cnrs.fr

A Mahler function is a solution, analytic in some neighborhood of the origin, of a linear difference equation associated with the Mahler operator $z\mapsto z^q$, where $q\geq 2$ is an integer. Understanding the nature of such functions at algebraic points of the complex open unit disc is an old number theoretical problem dating back to the pioneering works of Mahler in the late 1920s. In this talk, I will explain why it can be considered as totally solved now, after works of Ku. Nishioka, Philippon, Faverjon and the speaker.

Joint work with Colin Faverjon (France).

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July 14, 18:00 ~ 18:25 - Room B2

## A generalization of an integrability theorem of Darboux

### North Carolina State University, USA   -   iakogan@ncsu.edu

In Chapter I, Livre III of his monograph "Systèmes Orthogonaux'' Darboux stated three integrability theorems. They provide local existence and uniqueness of solutions to systems of first order Partial Differential Equations of the type: $\partial_{x_i} u_\alpha(x)=f^\alpha_i(x,u(x)), \quad i\in I_\alpha\subset\{1,\dots,n\}$ where $x_i$, $i=1,\dots, n$ are independent variables, and $u_\alpha$, $\alpha=1,\dots,m$ are dependent variables and $f^\alpha_i(x,u(x))$ are some given functions. For each dependent variable $u^\alpha$, the system prescribes partial derivatives in certain coordinate directions given by a subset $I_\alpha$. The data, near a given point $\bar x\in{\mathbb R}^n$, prescribe each of the unknown functions $u_\alpha$ along the affine subspace spanned by the coordinate vectors complimentary to the coordinate vector defined by indices $I_\alpha$.

Darboux's first theorem applies to determined systems, in which case $|I_\alpha|=1$ for all $\alpha$, while his second theorem is Frobenius' Theorem for complete systems, in which case $|I_\alpha|=n$ for all $\alpha$. The third theorem addresses the general situation where $I_\alpha$ are arbitrary subsets varying with $\alpha$. Under the appropriate integrability conditions, Darboux proved his third theorem in the cases $n=2$ and $n=3$. However, his argument does not appear to generalize in any easy manner to cases with more than three independent variables.

In the present work, we formulate and prove a theorem that generalizes Darboux's third theorem to systems of the form ${\mathbf r}_i(u_\alpha)\big|_x = f_i^\alpha (x, u(x)), \quad i\in I_\alpha\subset\{1,\dots,n\}$ where $\{{\mathbf r}_i\}_{i=1}^n$ is an arbitrary local frame of vector-fields near $\bar x$. Furthermore, the data for $u^\alpha$ can be prescribed along an arbitrary submanifold through $\bar x$ transversal to the subset of vector-fields $\{{\mathbf r}_i\,|\, i\in I_\alpha\}$. Our proof applies to any number of independent variables and uses a nonstandard application of Picard iteration. The approach requires only $C^1$ smoothness of the $f_i^\alpha$ and the initial data. We note that, in the analytic case, this result can be proved using Cartan-Kähler theorem.

Joint work with Michael Benfield, San Diego, USA and Helge Kristian Jenssen, Pennsylvania State University, USA.

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July 15, 14:30 ~ 15:20 - Room B2

## The Good, the Bad, and the Ugly: the Cartan algorithm for overdetermined PDE systems

### Jeanne Clelland

Cartan's theory of exterior differential systems provides tools for analyzing spaces of local solutions to systems of PDEs that don't fit nicely into any standard classification. In particular, the Cartan-Kahler Theorem -- which generalizes the Cauchy-Kowalewski Theorem for determined systems -- can often be used to compute the size of the local solution space, as long as the system itself and all initial data are assumed to be real analytic. This can be a powerful framework for analyzing the solution spaces to overdetermined PDE systems, including many that arise naturally in geometric contexts.

In this talk, we will illustrate the application of Cartan's algorithm to the problem of finding strong Beltrami fields with nonconstant proportionality factor, i.e., vector fields $\mathbf{u}$ on an open set $U \subset \mathbb{R}^3$ with the property that $\nabla \times \mathbf{u} = f \mathbf{u}, \qquad \nabla \cdot \mathbf{u} = 0$ for some nonconstant function $f:U \to \mathbb{R}$. This example illustrates both the power of the method and some significant challenges that may arise during its application.

Joint work with Taylor Klotz (University of Colorado, Boulder).

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July 15, 15:30 ~ 15:55 - Room B2

## Differential invariants for time-like curves and their applications to a conformally invariant variational problem.

### Politecnico di Torino, Italy   -   emilio.musso@polito.it

In this talk I will focus on local differential invariants of a time-like curve with respect to the group of the Lorentz conformal transformations. The invariant of lower-order is a differential form of degree four (the "conformal strain") which, in turns, can be used to define natural parameterizations. When the conformal strain doesn't possesses zeroes, one can integrate its fourth root along the curve. This provides the simplest conformally invariant variational problem for time-like curves. I will give some hint on how to find the critical curves and I will discuss the existence of closed critical curves.

References

- E. Musso, L. Nicolodi, “Quantization of the conformal arclength functional on space curves”, Communications in Analysis and Geometry, to appear ; arXiv:1501.04101 [math.DG].

- A. Dzhalilov, E. Musso, L. Nicolodi, “Conformal geometry of time-like curves in the (1+2)-dimensional Einstein universe”, Nonlinear Analyisis Sereies A : Theory, Methods and Applications, 2016, Vol 143 p.224-255; arXiv:1603.01035 [math.DG].

- O. Eshkobilov, E.Musso, A conformally invariant variational problem for time-like curves, ArXiv 1566997 [mathDG 22 may 2016].

Joint work with L. Nicolodi (University of Parma, Italy), A. Dzhalilov (TTPU, Taschkent, Uzbekistan) and O. Eshkobilov (University of Torino, Italy).

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July 15, 16:00 ~ 16:25 - Room B2

## On the computation of simple forms and regular solutions of linear difference systems

### University of Limoges; XLIM , France   -   thomas.cluzeau@unilim.fr

In this talk, I will present a new algorithm for transforming any first-order linear difference system with factorial series coefficients into a simple system. Such an algorithm can be seen as a first step towards the computation of regular solutions since in a previous work we have developed an algorithm for computing regular solutions of simple linear difference system. Moreover, computing a simple form can also be used to characterize the nature of the singularity at infinity. If the singularity is regular, we are then reduced to a system of the first-kind. I will also present a direct algorithm for computing a formal fundamental matrix of regular solutions of such first-kind linear difference systems which yields an alternative to the algorithm for computing regular solutions of simple systems in the case of a regular singularity. Finally, the algorithms developed have been implemented in Maple thanks to our new package for handling factorial series. The talk will be illustrated by examples computed using our implementation.

Joint work with Moulay Barkatou (University of Limoges; XLIM, France) and Carole El Bacha (Lebanese University, Faculty of Sciences II, Lebanon).

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July 15, 17:00 ~ 17:25 - Room B2

## Algebraic certificates of disconnectedness

### LAAS-CNRS University of Toulouse, France   -   henrion@laas.fr

Given two disjoint semialgebraic subsets of a given compact semialgebraic set, we describe an algorithm that certifies that the two subsets are disconnected, i.e. that there is no continuous trajectory connecting them. The certificate of disconnectedness is a polynomial satisfying specific positivity conditions. It is computed by solving a hierarchy of convex moment-sum-of-squares semidefinite programs obtained by applying infinite-dimensional convex duality on a transport equation satisfied by measures attached to trajectories.

Joint work with Mohab Safey El Din (Université Pierre et Marie Curie, Paris, France).

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July 15, 17:30 ~ 17:55 - Room B2

## Differential Galois Theory and non-integrability of planar polynomial vector fields

### Juan J. Morales-Ruiz

We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, as a corollary of a previous result with Ramis and Simó on Hamiltonian systems, it is proved that a necessary condition for the existence of a meromorphic first integral is that the identity component of the Galois group of the higher order variational equations around a particular solution must be abelian. We illustrate this theorem with several families of examples. A key point in these applications is to check wether a suitable primitive is or not elementary. Using a theorem by Liouville, the problem is equivalent to the existence of a rational solution of a certain first order linear equation, the Risch equation. This is a classical problem studied by Risch in 1969, and the solution is given by the Risch algorithm''. In this way we will point out the connection of the non integrablity of polynomial fields with some higher transcendent functions, like the error function.

Joint work with Primitivo B. Acosta-Humánez (Universidad Simón Bolivar, Colombia), J. Tomás Lázaro (Universitat Politècnica de Catalunya, Spain) and Chara Pantazi (Universitat Politècnica de Catalunya, Spain).

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July 15, 18:00 ~ 18:25 - Room B2

## Algorithmic proof for the transcendence of D-finite power series

### Inria, France   -   alin.bostan@inria.fr

Given a sequence represented by a linear recurrence with polynomial coefficients and sufficiently many initial terms, a natural question is whether the transcendence of its generating function can be decided algorithmically. The question is non trivial even for sequences satisfying a recurrence of first order. An algorithm due to Michael Singer is sufficient, in principle, to answer the general case. However, this algorithm suffers from too high a complexity to be effective in practice. We will present a recent method that we have used to treat a non-trivial combinatorial example. It reduces the question of transcendence to a (structured) linear algebra problem.

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## On Strongly Consistent Finite Difference Approximations to the Navier-Stokes Equations

### KAUST, KSA   -   dmitry.lyakhov@kaust.edu.sa

The finite difference method is widely used for solving partial differential equations in the computational sciences. The decisive factor for its successful application is the quality of the underlying finite difference approximations. In this contribution, we present a computer algebra assisted approach to generate appropriate finite difference approximations to systems of polynomially nonlinear partial differential equations on regular Cartesian grids. The generated approximations satisfy the major quality criterion -- strong consistency -- which implies the preservation of fundamental algebraic properties of the system at the discrete level. This criterion admits a verification algorithm. We apply our approach to the Navier-Stokes equations and construct strongly consistent approximations. Moreover, we construct two approximations which are not only strongly consistent but also fully conservative.

Joint work with Vladimir Gerdt (JINR, Russia) and Dominik Michels (KAUST, KSA).

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## Symbolic computation for operators with matrix coefficients

### Johannes Kepler University Linz, Austria   -   clemens.raab@jku.at

In order to facilitate symbolic computations with systems of linear functional equations an algebraic framework for such systems is needed for effective computations in corresponding rings of operators. Normal forms of operators are a key ingredient for that.

We generalize the recently developed tensor approach from scalar equations to the matrix case, by allowing noncommutative coefficients. The tensor approach is flexible enough to cover many operators, like integral operators, that do not fit the well-established framework of skew-polynomials. Noncommutative coefficients even allow to handle systems of generic size. Normal forms rely on a confluent reduction system.

Based on our implementation of tensor reduction systems, we implemented the ring of integro-differential operators with time-delay and we worked out normal forms for those operators. We use this to partly automatize certain computations related to differential time-delay systems, e.g. Artstein's reduction of differential time-delay control systems.

Joint work with Thomas Cluzeau (University of Limoges, France), Jamal Hossein Poor (RICAM, Austrian Academy of Sciences, Austria), Alban Quadrat (INRIA Lille - Nord Europe, France) and Georg Regensburger (Johannes Kepler University Linz, Austria).

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## Discrete Moving Frames and Noether’s Finite Difference Conservation Laws. Euler’s Elastica.

### University of Kent, United Kingdom   -   arer2@kent.ac.uk

Discrete moving frames have the potential to have a large spectrum of applications as well as leading to improvement regarding computations. Further, the theory of discrete moving frames provides an useful tool when solving variational problems or in the study of integrable systems. In this poster I will consider the discrete variational problem analogue to the minimisation of the integral of the curvature squared. The solutions of this equation are commonly known as Euler’s elastica. I will show that the Euler-Lagrange difference equations and the Noether’s difference conservation laws can be written in terms of the invariants of the action and a discrete moving frame and that the appearance of the moving frame in the expression for the conservation laws makes explicit the equivariance of the frame under the group action. I will exhibit how this formulation can allow one, to solve for the solutions in term of the original variables. By matching the use of the smooth and discrete frames, for the smooth and discrete problems respectively, I will show that it is possible to design approximations which have matching conservation laws.

Joint work with Elizabeth Mansfield (University of Kent, UK).

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FoCM 2017, based on a nodethirtythree design.