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… 
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