My recent works relate to a new “parametrization” of “good” convex polygons of the lattice of integers $ \mathbb {Z} ^ 2 $, involving chains of affine references, via positive chains (particular affine applications), of which the main objective would be to obtain an efficient algorithm allowing to determine the whole convex polygons of minimal areas, modulo the affine special group.

The study of positive chains also leads to combinatorial results making use Catalan numbers via considerations bordering on theoretical computer science, with the study of formal languages, monoid presentations by generators and relations of the modular group $ PSL_2 (\mathbb {Z}) $ (as a monoid) and combinatorial objects such as binary trees.

I am also interested in algebraic integers whose Galois conjugates have particular properties with respect to the unit circle. I study in particular new constructions, limit properties, root interlacing phenomena, (…), minimal polynomials of these algebraic integers, namely: cyclotomic polynomials, Pisot polynomials, Salem polynomials, expansive polynomials.