What is the degree of a smooth hypersurface?

Antonio Lerario and Michele Stecconi

Journal of Singularities
volume 23 (2021), 205-235

Received: 22 April 2021. Received in revised form: 16 November 2021.

DOI: 10.5427/jsing.2021.23l

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Abstract:

We deal with the problem of the algebraic approximation of type-W singularities of smooth functions on a closed n-disk, namely the set of points in the disk where the jet extension of the function meets a given semialgebraic subset W of the jet space; examples of sets arising in this way are the zero set of a function, or the set of its critical points.

We prove that the type-W singularity defined by a smooth function, satisfying a transversality condition, is isotopic to the one defined by a polynomial whose degree is explicitly bounded in terms of the distance of the original function from the set of functions which do not satisfy the transversality condition. Ultimately, the bound depends on the second derivatives of the jet of the function. The estimate on the degree of the approximating polynomial implies an estimate on the Betti numbers of the original singularity. Using more refined and recently developed tools, we prove a second bound, directly for Betti numbers, that is of lower order than that implied by the previous one, in that it depends only on the first derivatives of the jet.

These results specialize to the case of a smooth compact hypersurface, resulting in a control of the minimal degree of its algebraic realization (from which the title of the paper) and of its Betti numbers. As a corollary we prove an upper bound on the number of isotopy classes of compact hypersurfaces satisfying a certain quantitative transversality condition. Moreover, we show that in this case the second estimate on the Betti numbers is asymptotically sharp. Finally, we relate the two estimates - the one for the degree of a polynomial realization and the one for the Betti numbers - with the geometric data of the hypersurface, independent from its defining equation, showing that the bounds can be given in terms of the reach and the diameter.

Author(s) information:

Antonio Lerario
SISSA
Via Bonomea 265
34136 Trieste, Italy
email: lerario@sissa.it

Michele Stecconi
Laboratoire de Mathématiques Jean Leray
2 rue de la Houssinière
44300, Nantes, France
email: stecconi.michele@gmail.com