On some combinatorial formulae coming from Hessian Topology

TL;DR

This paper explores combinatorial identities derived from Hessian topology that aid in understanding the topological properties of spaces related to hyperbolic polynomials, highlighting the interplay between combinatorics and geometry.

Contribution

It introduces new binomial coefficient identities obtained through various combinatorial methods, relevant for topological analysis of hyperbolic polynomial graphs.

Findings

Identified key binomial identities useful in Hessian topology

Proved identities using multiple combinatorial approaches

Applied identities to topological properties of hyperbolic polynomial spaces

Abstract

The interaction between combinatorics and algebraic and differential geometry is very strong. While researching a problem of Hessian topology, we came across a series of identities of binomial coefficients, which are useful for proving a topological property of certain spaces whose elements are graphs of a class of hyperbolic polynomials. These identities are proven by different methods in combinatorics.