The AROMATH seminar will takes place typically every second Wednesday at 14:00 (FR), except for a few deviations.

The presentations will typically take place at Inria Sophia Antipolis, Byron Blanc 106, and online at https://cutt.ly/aromath.

Click on a talk to see the abstract.

## Events in January–June 2021

**-**Pablo Gonzalez Mazon - Center sets and the Carathéodory number of real algebraic varieties–

### Pablo Gonzalez Mazon - Center sets and the Carathéodory number of real algebraic varieties

The so-called "Center point theorem" is a classical result in combinatorial geometry, that generalizes the notion of the median of a finite subset of $\mathbb{R}$ to point-sets in $\mathbb{R}^n$. A natural question is whether similar statements hold not only for (closed) halfspaces, but for sets defined by a single polynomial inequality in arbitrary degree. The answer yields to the definition of "center sets", which further extend the concept of a center point, and settle questions about their size. In particular, an upper bound for the size of a minimal center set is given by the Carathéodory number of the Veronese embedding of $\mathbb{R}^n$, establishing an interesting connection between combinatorial geometry and convex algebraic geometry. Further, we provide a sharp bound for the minimal size of a center set in $\mathbb{R}^2$ when the degree of the polynomials involved is at most 2. Additionally, given a smooth, compact, real algebraic variety X, we explore the relationship between its Carathéodory number and local algebraic properties of the associated dual variety X*.

**-**Sebastian debus (University of the Arctic in Tromso) - Reflection groups and cones of sums of squares–

### Sebastian debus (University of the Arctic in Tromso) - Reflection groups and cones of sums of squares

We consider cones of real forms which are sums of squares forms and invariant by a (finite) reflection group. We show how the representation theory of these groups allows to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the An, Bn, and Dn case where we use so called higher Specht polynomials (Ariki, Terasoma, Yamada 1997)to give a uniform description of these cones. These descriptions allow us, for example, to study the connection of these cones to non-negative forms. In particular, we give a new proof of a result by Harris (1999) who showed that every non-negative ternary even symmetric octic form is a sum of squares.Joint work with Cordian Riener (https://arxiv.org/pdf/2011.09997.pdf)

**-**Alban Quadrat (Inria Paris Ouragan) - On the general solutions of a rank factorization problem arising in vibration analysis–

### Alban Quadrat (Inria Paris Ouragan) - On the general solutions of a rank factorization problem arising in vibration analysis

Given a field K, r matrices D_1, …, D_r \in K^{n \times n} and a matrix M \in K^{n \times m}, in this talk, we shall study the problem of factoring M as followsM=\sum_{i=1}^r D_i u v_i,where u \in K^{n \times 1} and v_i \in K^{1 \times m} for i=1, …, r.This rank factorization problem arises in modulation-based mechanical models studied in gearbox vibration analysis. It amounts to solving a family of bilinear polynomial systems.In this talk, using module theory and homological algebra methods, we shall characterize the set of solutions of this rank factorization problem.The results will be illustrated with explicit examples computed by the package*CapAndHomalg*(*GAP)*developed by Mohamed Barakat (Univ. Siegen) and his collaborators.This work was done in collaboration with Yacine Bouzidi, Axel Barrau (Safran Tech), Roudy Dagher (Inria Lille) and Elisa Hubert (LASPI, Univ. Lyon).