José Fernando Galván - FB Mathematik und Statistik

Universität Konstanz
Fachbereich
Mathematik und Statistik
Schwerpunkt
Reelle Geometrie und Algebra
Einladung
Im Oberseminar Reelle Geometrie und Algebra hält
José Fernando Galván
(University Complutense de Madrid)
am Freitag, 24.10.2014, einen Vortrag zum Thema:
On the set of points at infinity of a polynomial
image of Rn (joint work with Carlos Ueno)
Der Vortrag findet um 13:30 Uhr in F426 statt.
Alle Interessenten sind herzlich eingeladen.
The study of polynomial and regular images of Rn has been
developed during the last 25 years. In our opinion we are
still very far from obtaining a complete solution to this problem. Between the most relevant results we point out: the
full characterizations of the 1-dimensional polynomial and
regular images of Rn , the connexion of the set of points at
infinity of a polynomial image (which is the main objective
of this seminar) or the results concerning the representation
of convex polyhedra and their interiors (as regular images)
or their complements and the corresponding closures (as
polynomial images). The representation of the open quadrant x>0,y>0 as a polynomial images of R2 has been a
crucial step for the representation of further families. In
this seminar we will sketch the proof of the connexion of
the set of points at infinity of a semialgebraic set S that is
the image of a polynomial map form Rn to Rm . The proof
involves important classical results from resolution of the
indeterminancy of rational maps, complex algebraic geometry and algebraic topology. Similar techniques (which have
inspired us) were used before in some Jelonek’s works developed in the 1990s, where he studied the set of points at
which a polynomial map is not proper. The connexion result
of the sets of points at infinity is no longer true for general
regular maps from Rn to Rm , although it still holds for an
ample family of regular maps that we call quasi-polynomial
maps. We will also present some enlightening examples for
the general regular case.
As far as we know the general problem of determining which
semialgebraic subsets of Rm are either polynomial or regular
images of Rn was initially proposed by J.M. Gamboa in the
1990 congress ”Réelle Algebraische Geometrie” celebrated
in Oberwolfach. The results mentioned above have been
developed together with either J.M. Gamboa or C. Ueno.