## Section: Partnerships and Cooperations

### International Initiatives

#### Participation in Other International Programs

##### PICS project

Participant : Laurent Busé.

We participate to a bilateral collaboration between France and Spain which is supported as a PICS from CNRS. This project, titled *Diophantine Geometry and Computer Algebra*, aims at exploring interactions between diophantine geometry and computer algebra by stimulating collaborations between experts in both domains. The research program focuses on five particular topics : toric varieties and height, equidistribution, Diophantine geometry and complexity, factorization of multivariate polynomials by means of toric geometry and study of singularities of toric parameterizations.

The Spanish partner is the University of Barcelona, with participants J. Burgos, C. D'Andrea, Martin Sombra, and the French partners are the university of Caen, with participants F. Amoroso and M. Weimann, the University of Paris 6, with participants M. Chardin and P. Philippon and the Inria project-team AROMATH, with participant L. Busé.

##### SYRAM project

Participants : Laurent Busé, Bernard Mourrain, André Galligo.

**Title:** Geometry of SYzygies of RAtional Maps with applications to geometric modeling (SYRAM)

We coordinate a research project which is funded by the regional program Math-AmSud for two years : 2015-2016. This project is composed by research teams from Argentina, Universidad de Buenos Aires (Nicolás Botbol, Alicia Dickenstein), Brazil, Universidade Federal de Rio de Janeiro, de Pernambuco e de Sergipe (Sayed Hamid Hassanzadeh, Aron Simis) and France, Institut de Mathématiques de Jussieu (Marc Chardin) and the Inria project-team AROMATH.

The study of rational maps is of theoretical interest in algebraic geometry and commutative algebra, and of practical importance in geometric modeling. This research proposal focuses on rational maps in low dimension, typically parameterizations of curves and surfaces embedded in the projective space of dimension three, but also dominant rational maps in dimension two and three. The two main objectives amount to unravel geometric properties of these rational maps from the syzygies of their projective coordinates. The first one aims at extending and generalizing the determination of the closed image of a rational map, as well as its geometric features, whereas the second one will focus on the study of dominant rational maps, in particular on the characterization of those that are generically one-to-one.