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 ternary quartics, i.e. degree 4 homogeneous polynomials in 3 variables, is, for instance, relevant in determining biomarkers for white matter from diffusion MRI.
We characterize a generating set of rational invariants of the orthogonal group acting on even degree forms by their restriction on a slice. These restrictions are invariant under the octahedral group and their explicit formulae are given compactly in terms of equivariant maps. The invariants of the orthogonal group can then be obtained in an explicit way, but their numerical evaluation can be achieved more robustly using their restrictions. The exhibited set of generators futhermore allows us to solve the inverse problem and the rewriting.
Central in obtaining the invariants for higher degree forms is the preliminary construction, with explicit formulae, for a basis of harmonic polynomials with octahedral symmetry, dif- ferent, though related, to cubic harmonics.
This is joint work with Paul Görlach (now at MPI Leipzig), in a joint project with Téo Papadopoulo (Inria Méditerranée).[-]

D-finite functions play a prominent role in computer algebra because they are well suited for representation in a symbolic software system, and because they include many functions of interest, such as special functions, orthogonal polynomials, generating functions from combinatorics, etc. Whenever one wishes to study the integral or the sum of a D-finite function, the method of creative telescoping may be applied. This method has been systematically introduced by Zeilberger in the 1990s, and since then has found applications in various different domains. In this lecture, we explain the underlying theory, review some of the history and talk about some recent developments in this area.[-]

Matrices whose coefficients are univariate polynomials over a field are a basic mathematical object which arises at the core of fundamental algorithms in computer algebra: sparse or structured linear system solving, rational approximation or interpolation, division with remainder for bivariate polynomials, etc. After presenting this context, we will give an overview of recent progress on efficient computations with such matrices. Next, we will show how these results have been exploited to improve complexity bounds for a selection of problems which, interestingly, do not necessarily involve polynomial matrices a priori: computing the characteristic polynomial of a scalar matrix, performing modular composition of univariate polynomials, changing the monomial order for multivariate Gröbner bases.[-]

D-finite functions play a prominent role in computer algebra because they are well suited for representation in a symbolic software system, and because they include many functions of interest, such as special functions, orthogonal polynomials, generating functions from combinatorics, etc. Whenever one wishes to study the integral or the sum of a D-finite function, the method of creative telescoping may be applied. This method has been systematically introduced by Zeilberger in the 1990s, and since then has found applications in various different domains. In this lecture, we explain the underlying theory, review some of the history and talk about some recent developments in this area.[-]

Matrices whose coefficients are univariate polynomials over a field are a basic mathematical object which arises at the core of fundamental algorithms in computer algebra: sparse or structured linear system solving, rational approximation or interpolation, division with remainder for bivariate polynomials, etc. After presenting this context, we will give an overview of recent progress on efficient computations with such matrices. Next, we will show how these results have been exploited to improve complexity bounds for a selection of problems which, interestingly, do not necessarily involve polynomial matrices a priori: computing the characteristic polynomial of a scalar matrix, performing modular composition of univariate polynomials, changing the monomial order for multivariate Gröbner bases.[-]

Robotic design involves modeling the behavior of a robot mechanism when it moves along potential paths set by the users. In this lecture, we will first give an overview of different approaches to design a set of kinematic equations associated with a robot mechanism. In particular, these equations can be used to solve the forward and the backward kinematics problems associated with a robot mechanism or to model its singularity locus. Then we will review methods to solve those equations, and notably methods to draw with guarantees the real solutions of an under-constrained system of equations modeling the singularities of a robot.[-]

Robotic design involves modeling the behavior of a robot mechanism when it moves along potential paths set by the users. In this lecture, we will first give an overview of different approaches to design a set of kinematic equations associated with a robot mechanism. In particular, these equations can be used to solve the forward and the backward kinematics problems associated with a robot mechanism or to model its singularity locus. Then we will review methods to solve those equations, and notably methods to draw with guarantees the real solutions of an under-constrained system of equations modeling the singularities of a robot.[-]