Note sur la règle des signes de Descartes
On Brauer groups of real Enriques surfaces.
On the finiteness of Pythagoras numbers of real meromorphic functions
We consider the 17th Hilbert Problem for global real analytic functions in a modified form that involves infinite sums of squares. Then we prove a local-global principle for a real global analytic function to be a sum of squares of global real meromorphic functions. We deduce that an affirmative solution to the 17th Hilbert Problem for global real analytic functions implies the finiteness of the Pythagoras number of the field of global real meromorphic functions, hence that of the field of real...
Overdetermined Strata in General Families of Polynomials
2000 Mathematics Subject Classification: 12D10.In the paper we give different examples of overdetermined strata.
Positive polynomials and hyperdeterminants
Let F be a homogeneous real polynomial of even degree in any number of variables. We consider the problem of giving explicit conditions on the coefficients so that F is positive definite or positive semi-definite. In this note we produce a necessary condition for positivity, and a sufficient condition for non-negativity, in terms of positivity or semi-positivity of a one-variable characteristic polynomial of F. Also, we revisit the known sufficient condition in terms of Hankel matrices.
Pseudo-real principal Higgs bundles on compact Kähler manifolds
Let be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let be a connected complex reductive affine algebraic group equipped with a real form . We define pseudo-real principal -bundles on . These are generalizations of real algebraic principal -bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal -bundles. Their relationships with the usual stable, semistable...
Real Schubert calculus: polynomial systems and a conjecture of Shapiro and Shapiro.
Real theta-characteristics on real projective curves.
Rokhlin conjecture and quotients of complex surfaces by complex conjugation.
Strict uniformization of real algebraic curves and global real analytic coordinates on real Teichmüller spaces.
We construct a global system of real analytic coordinates on the real Teichmüller space of a compact real algebraic curve X, using so-called strict uniformization of the real algebraic curve X. A global coordinate system is then obtained via real quasiconformal deformations of the Kleinian subgroup of PGL2(R) obtained as a group of covering transformations of a strict uniformization of X.
Sums of powers in rings and the real holomorphy ring.
The Gaussian measure on algebraic varieties
We prove that the ring ℝ[M] of all polynomials defined on a real algebraic variety is dense in the Hilbert space , where dμ denotes the volume form of M and the Gaussian measure on M.
The Maslov dequantization, idempotent and tropical mathematics: a brief introduction.
The Nullstellensatz for real coherent analytic surfaces.
The special Schubert calculus is real.
The tropical Grassmannian.
Théorèmes de finitude pour les variétés pfaffiennes
On introduit, dans ce travail, une hypothèse sur le spiralement d’une feuille d’un feuilletage analytique réel de codimension un (hypersurface pfaffienne). On en tire des résultats très généraux de finitude du type de Khovanskii. Des exemples précis montrent la généralité de ces hypersurfaces pfaffiennes. Une description complété des bouts de telles variétés en dimension trois est donnée.
Ueber die Vieltheiligkeit der ebenen algebraischen Curven