Inspiration of a real algebraic hypersurface.
(Épaississement d’une hypersurface algébrique réelle.)

*(French)*Zbl 0788.14051The 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.

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)

##### Keywords:

deformation of a real surface; real closed field; perturbation; number of connected components of a hypersurface
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\textit{M. Coste}, Proc. Japan Acad., Ser. A 68, No. 7, 175--180 (1992; Zbl 0788.14051)

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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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