Parusinski's “Key Lemma” via algebraic geometry.
Polynomial inequalities on algebraic sets
We give an estimate of Siciak’s extremal function for compact subsets of algebraic varieties in (resp. ). As an application we obtain Bernstein-Walsh and tangential Markov type inequalities for (the traces of) polynomials on algebraic sets.
Polynomial mappings form products of algebraic sets into spheres.
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.
Préhistoire de la géométrie algébrique réelle : de Descartes à Tarski
Projective modules over real affine algebras.
Prolongements en fonctions algébriquement constructibles.
In this paper we consider the following question: Let S be a semialgebraic subset of a real algebraic set V, and let φ: S → Z be a function on S. Is φ the restriction of an algebraically constructible function on V, i.e. a sum of signs of polynomials on V? We give an effective method to answer this question when φ(S) ⊂ {-1,1} or dim S ≤ 2 or S is basic.
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...