## Section: New Results

### Algebraic algorithms for geometric computing

#### Resultant of an equivariant polynomial system with respect to the symmetric group

Participants : Laurent Busé, Anna Karasoulou.

Given a system of n homogeneous polynomials in n variables which is equivariant with respect to the canonical actions of the symmetric group of n symbols on the variables and on the polynomials, it is proved that its resultant can be decomposed into a product of several smaller resultants that are given in terms of some divided differences. As an application, we obtain a decomposition formula for the discriminant of a multivariate homogeneous symmetric polynomial.

This work is submitted for publication [14] .

#### Delaunay Mesh Generation of NURBS Surfaces

Participant : Laurent Busé.

We introduce a method for isotropic triangle meshing of NURBS surfaces. Based on Delaunay filtering and refinement, our approach departs from previous work by meshing in embedding space instead of parametric space. The meshing engine relies upon a novel line/surface intersection test, based on the matrix-based implicit representation of NURBS surfaces and numerical methods in linear algebra such as singular value and eigenvalue decompositions. A careful treatment of degenerate cases makes our approach robust to intersection points with multiple pre-images. In addition to ensure both approximation accuracy and mesh quality, our approach is seamless as it does not depend on the initial decomposition into NURBS patches, and is oblivious to the parameterization of the patches. Removing such dependencies provides us with a means to reliably mesh across patches with greater control over mesh sizing and shape of the elements.

This work was done in collaboration with Jingjing Shen and Neil Dodgson from Cambridge University and Pierre Alliez from TITANE.

#### Toric Border Basis

Participant : Bernard Mourrain.

In [11] , we extend the theory and the algorithms of Border bases to systems of Laurent polynomial equations, defining “toric” roots. Instead of introducing new variables and new relations to saturate by the variable inverses, we propose a more efficient approach which works directly with the variables and their inverse. We show that the commutation relations and the inversion relations characterize toric border bases. We explicitly describe the first syzygy module associated to a toric border basis in terms of these relations. Finally, a new border basis algorithm for Laurent polynomials is described and a proof of its termination is given for zero-dimensional toric ideals.

Joint work with Philippe Trébuchet (LIP6 - UPMC).

#### Border Basis relaxation for polynomial optimization

Participants : Marta Abril-Bucero, Bernard Mourrain.

A relaxation method based on border basis reduction which improves the efficiency of Lasserre's approach is proposed to compute the optimum of a polynomial function on a basic closed semi-algebraic set. A new stopping criterion is given to detect when the relaxation sequence reaches the minimum, using a sparse flat extension criterion. We also provide a new algorithm to reconstruct a finite sum of weighted Dirac measures from a truncated sequence of moments, which can be applied to other sparse reconstruction problems. As an application, we obtain a new algorithm to compute zero-dimensional minimizer ideals and the minimizer points or zero-dimensional G-radical ideals. Experimentations show the impact of this new method on significant benchmarks. See [12] .

#### Flat extensions in $*$-algebra

Participant : Bernard Mourrain.

The objective of this work is to develop a flat extension characterization on moment matrices in the non-commutative case. We give a flat extension theorem for positive linear functionals on $\ast $-algebras. The theorem is applied to truncated moment problems on cylinder sets, on matrices of polynomials and on enveloping algebras of Lie algebras. See [17] .

Joint work with Konrad Schmüdgen, University of Leipzig, Germany.