zbMATH — the first resource for mathematics

Inspiration of a real algebraic hypersurface. (Épaississement d’une hypersurface algébrique réelle.) (French) Zbl 0788.14051
The author gives a nice proof (working also in the case of a real closed field) ot a result of Hironaka on the existence of a deformation of a real surface with equation \(P=0\), such that it has the same degree as \(P\) and moreover all the set \(P=0\) is contained in the closure of the deformation.
This does not happen, for instance, if one deforms the curve \[ P(x,y)=x[(x-1)^ 2(x+1)^ 2+Y^ 2)]=0 \] with \(P=\varepsilon\). If \(d=\deg P\), Hironaka’s proof uses polynomials of the form: \[ P^{(0)}=P,\quad P^{(I+1)}=P^{(I)}+\sum^ n_{i=1}t_ i^{(I+1)}{\partial P^{(I)}\over\partial X_ i} \] with \(I=1,\dots,d\).
In the case of a real closed field \(R\), the indeterminates are choosen to be infinitesimal with respect to \(R\). Suppose \(P^{(0)}\) to be monic with respect to the variable \(x_ 1\): then the first perturbation by an infinitesimal \(\varepsilon_ 1\) separates a simple root from a multiple one. Then a generic perturbation is given by other infinitesimals without loosing the simple root. At the end of the process one can eliminate the infinitesimals and get a deformation with parameters in an open set of the space of polynomials of degree \(d\).
A bound on the number of connected components of a hypersurface of degree \(d\) is proved as a consequence of the main theorem. The author gets similar results also for the case of projective hypersurfaces.
Reviewer: F.Broglia (Pisa)

14P05 Real algebraic sets
14J70 Hypersurfaces and algebraic geometry
Full Text: DOI
[1] R. Benedetti and J-J. Risler: Real algebraic and semi-algebraic sets. Hermann, Paris (1990). · Zbl 0694.14006
[2] J. Bochnak, M. Coste and M-F. Roy: G6ometrie algebrique reelle. Springer, Berlin, Heidelberg (1987).
[3] I. Fary: Cohomologie des varietes algebriques. Ann. Math., 65, 21 - 73 (1957). JSTOR: · Zbl 0082.36504 · doi:10.2307/1969665 · links.jstor.org
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.