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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)

MSC:
 14P05 Real algebraic sets 14J70 Hypersurfaces and algebraic geometry
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References:
 [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
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