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On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real Closed Fields

Sven Wagner (2010)

Annales de la faculté des sciences de Toulouse Mathématiques

We will prove that the Pierce-Birkhoff Conjecture holds for non-singular two-dimensional affine real algebraic varieties over real closed fields, i.e., if W is such a variety, then every piecewise polynomial function on W can be written as suprema of infima of polynomial functions on W . More precisely, we will give a proof of the so-called Connectedness Conjecture for the coordinate rings of such varieties, which implies the Pierce-Birkhoff Conjecture.

On the rings of formal solutions of polynomial differential equations

Maria-Angeles Zurro (1998)

Banach Center Publications

The paper establishes the basic algebraic theory for the Gevrey rings. We prove the Hensel lemma, the Artin approximation theorem and the Weierstrass-Hironaka division theorem for them. We introduce a family of norms and we look at them as a family of analytic functions defined on some semialgebraic sets. This allows us to study the analytic and algebraic properties of this rings.

On the Weierstrass division.

Łojasiewicz, Stanisław, Maszczyk, Tomasz, Rusek, Kamil (2001)

Zeszyty Naukowe Uniwersytetu Jagiellońskiego. Universitatis Iagellonicae Acta Mathematica

Ordered fields.

Francis RAYNER (1975/1976)

Seminaire de Théorie des Nombres de Bordeaux

Points rationnels et méthode de Chabauty elliptique

Sylvain Duquesne (2003)

Journal de théorie des nombres de Bordeaux

La méthode de Chabauty elliptique permet de calculer les points rationnels sur une courbe définie sur un corps de nombres lorsque le théorème de Chabauty ne s’applique pas, c’est à dire lorsque le rang de la jacobienne est supérieur au genre de la courbe. Nous exposons cette méthode et nous la généralisons dans de nouveaux cas en écrivant une version explicite du théorème de préparation de Weierstrass en 2 variables. En particulier nous calculons tous les points rationnels d’une courbe de genre...

Pointwise convergence fails to be strict

Ján Borsík, Roman Frič (1998)

Czechoslovak Mathematical Journal

It is known that the ring B ( ℝ ) of all Baire functions carrying the pointwise convergence yields a sequential completion of the ring C ( ℝ ) of all continuous functions. We investigate various sequential convergences related to the pointwise convergence and the process of completion of C ( ℝ ) . In particular, we prove that the pointwise convergence fails to be strict and prove the existence of the categorical ring completion of C ( ℝ ) which differs from B ( ℝ ) .

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.

Currently displaying 141 – 160 of 292