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Parusinski's “Key Lemma” via algebraic geometry.

Reichstein, Z., Youssin, B. (1999)

Electronic Research Announcements of the American Mathematical Society [electronic only]

Polynomial inequalities on algebraic sets

M. Baran, W. Pleśniak (2000)

Studia Mathematica

We give an estimate of Siciak’s extremal function for compact subsets of algebraic varieties in $ℂ^{n}$ (resp. $ℝ^{n}$ ). 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.

J., Kucharz, W. Bochnak (1991)

Journal für die reine und angewandte Mathematik

Positive polynomials and hyperdeterminants

Fernando Cukierman (2007)

Collectanea Mathematica

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

Hourya Sinaceur (1991)

Cahiers du séminaire d'histoire des mathématiques

Projective modules over real affine algebras.

R. Parimala, M. Ojanguren (1990)

Mathematische Annalen

Prolongements en fonctions algébriquement constructibles.

Isabelle Bonnard-Doré (2004)

Revista Matemática Complutense

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

Indranil Biswas, Oscar García-Prada, Jacques Hurtubise (2014)

Annales de l’institut Fourier

Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_{G}$ . We define pseudo-real principal $G$ -bundles on $X$ . These are generalizations of real algebraic principal $G$ -bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$ -bundles. Their relationships with the usual stable, semistable...

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