On Euler polynomials for projective hypersurfaces

TL;DR

This paper introduces Euler polynomials that encode all Chern numbers of smooth hypersurfaces in projective space, providing a unified polynomial framework to recover these invariants from a single polynomial.

Contribution

It defines Euler polynomials for each positive integer n and demonstrates they encapsulate all Chern numbers of hypersurfaces in projective space, extending to hypersurfaces in smooth varieties.

Findings

Euler polynomial $\\mathscr{E}_n(t)$ encodes all Chern numbers for hypersurfaces in $\mathbb{P}^n$

All Chern classes of hypersurfaces in a smooth variety can be recovered from the top Chern class

The polynomial framework simplifies the computation of numerical invariants of hypersurfaces

Abstract

For every positive integer $n\in \mathbb{Z}_+$ we define an `Euler polynomial' $\mathscr{E}_n(t)\in \mathbb{Z}[t]$, 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 $\mathscr{E}_n(t)$. More generally, we show that all Chern classes of hypersurfaces in a smooth variety may be recovered from its top Chern class.