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).[-]

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.[-]

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.[-]

I will start from reviewing Gröbner bases and their connection to polynomial system solving. The problem of solving a polynomial system of equations over a finite field has relevant applications to cryptography and coding theory. For many of these applications, being able to estimate the complexity of computing a Gröbner basis is crucial. With these applications in mind, I will review linear-algebra based algorithms, which are currently the most efficient algorithms available to compute Gröbner bases. I will define and compare several invariants, that were introduced with the goal of providing an estimate on the complexity of computing a Gröbner basis, including the solving degree, the degree of regularity, and the last fall degree. Concrete examples will complement the theoretical discussion.[-]

I will start from reviewing Gröbner bases and their connection to polynomial system solving. The problem of solving a polynomial system of equations over a finite field has relevant applications to cryptography and coding theory. For many of these applications, being able to estimate the complexity of computing a Gröbner basis is crucial. With these applications in mind, I will review linear-algebra based algorithms, which are currently the most efficient algorithms available to compute Gröbner bases. I will define and compare several invariants, that were introduced with the goal of providing an estimate on the complexity of computing a Gröbner basis, including the solving degree, the degree of regularity, and the last fall degree. Concrete examples will complement the theoretical discussion.[-]