On Euler polynomials for projective hypersurfaces

Abstract

For every positive integer $n\in {\mathbb{Z}}_{+}$ we define an `Euler polynomial' ${\mathcal{E}}_n(t)\in \mathbb{Z}[t]$, and observe that for a fixed $n$ all Chern numbers (as well as other numerical invariants) of all smooth hypersurfaces in ${\mathbb{P}}^n$ may be recovered from the single polynomial ${\mathcal{E}}_n(t)$. More generally, we show that all Chern classes of hypersurfaces in a smooth variety may be recovered from its top Chern class.