What is the degree of a smooth hypersurface?

@article{Lerrio2021WhatIT,
  title={What is the degree of a smooth hypersurface?},
  author={Antonio Marcondes Ler{\'a}rio and Michele Stecconi},
  journal={Journal of Singularities},
  year={2021}
}
Let $D$ be a disk in $\mathbb{R}^n$ and $f\in C^{r+2}(D, \mathbb{R}^k)$. We deal with the problem of the algebraic approximation of the set $j^{r}f^{-1}(W)$ consisting of the set of points in the disk $D$ where the $r$-th jet extension of $f$ meets a given semialgebraic set $W\subset J^{r}(D, \mathbb{R}^k).$ Examples of sets arising in this way are the zero set of $f$, or the set of its critical points. Under some transversality conditions, we prove that $f$ can be approximated with a… 
1 Citations

Figures from this paper

Higher derivatives of functions vanishing on a given set

Let f : B → R be a d + 1 times continuously differentiable function on the unit ball B, with max z∈Bn‖f(z)‖ = 1. A well-known fact is that if f vanishes on a set Z ⊂ B with a non-empty interior, then

References

SHOWING 1-10 OF 26 REFERENCES

Maximal and Typical Topology of Real Polynomial Singularities.

Given a polynomial map $\psi:S^m\to \mathbb{R}^k$ with components of degree $d$, we investigate the structure of the semialgebraic set $Z\subseteq S^m$ consisting of those points where $\psi$ and its

The set of zeroes of an “almost polynomial” function

Let / be a smooth function on the unit «-dimensional ball, with the C°-norm, equal" to one. We prove that if for some k > 2, the norm of the kth derivative of/is bounded by 2~*_1, then the set of

Elliptic Partial Differential Equations of Second Order

We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations

On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets

  • S. Basu
  • Mathematics, Computer Science
    Discret. Comput. Geom.
  • 1999
TLDR
It is proven that the sum of the Betti numbers of S is bounded by sk' 2O(k2 m4) in case the total number of monomials occurring in the polynomials in $ {\cal P} \cup \{Q\}$ is m.

NASH’S WORK IN ALGEBRAIC GEOMETRY

This article is a survey of Nash’s contributions to algebraic geometry, focusing on the topology of real algebraic sets and on arc spaces of singularities. Nash wrote two papers in algebraic

On the Betti numbers of real varieties

PROOF. Approximate fi, * *, fm by real polynomials F1, * , Fm of the same degrees whose coefficients are algebraically independent. Now consider the variety Vc in the complex Cartesian space

3264 and All That: A Second Course in Algebraic Geometry

Introduction 1. Introducing the Chow ring 2. First examples 3. Introduction to Grassmannians and lines in P3 4. Grassmannians in general 5. Chern classes 6. Lines on hypersurfaces 7. Singular

2 ,

Since 2001, we have observed the central region of our Galaxy with the near-infrared (J, H, and Ks) camera SIRIUS and the 1.4 m telescope IRSF. Here I present the results about the infrared

Real algebraic geometry

1. Ordered Fields, Real Closed Fields.- 2. Semi-algebraic Sets.- 3. Real Algebraic Varieties.- 4. Real Algebra.- 5. The Tarski-Seidenberg Principle as a Transfer Tool.- 6. Hilbert's 17th Problem.