The Stratified Spaces of Real Polynomials & Trajectory Spaces of Traversing Flows

TL;DR

This paper studies the combinatorial structure of traversally generic flows on manifolds with boundary, linking it to the stratification of real polynomial spaces and revealing cellular structures that describe trajectory behaviors.

Contribution

It introduces universal posets capturing boundary tangency patterns and connects them to the stratification of real polynomial spaces, advancing Morse theory on manifolds with boundary.

Findings

Posets $oldsymbol{\Omega^ullet_{' extless n]}}$ encode boundary tangency combinatorics.

Cellular stratification of polynomial spaces $oldsymbol{\mathcal{P}^d}$ by poset elements.

Local structure of trajectory spaces $oldsymbol{\mathcal{T}(v)}$ characterized by these combinatorial models.

Abstract

This paper is the third in a series that researches the Morse Theory, gradient flows, concavity and complexity on smooth compact manifolds with boundary. Employing the local analytic models from \cite{K2}, for \emph{traversally generic flows} on $(n+1)$-manifolds $X$, we embark on a detailed and somewhat tedious study of universal combinatorics of their tangency patterns with respect to the boundary $\d X$. This combinatorics is captured by a universal poset $\Omega^\bullet_{'\langle n]}$ which depends only on the dimension of $X$. It is intimately linked with the combinatorial patterns of real divisors of real polynomials in one variable of degrees which do not exceed $2(n+1)$. Such patterns are elements of another natural poset $\Omega_{\langle 2n+2]}$ that describes the ways in which the real roots merge, divide, appear, and disappear under deformations of real polynomials. The space of real degree $d$ polynomials $\mathcal P^d$ is stratified so that its pure strata are cells, labelled by the elements of the poset $\Omega_{\langle d]}$. This cellular structure in $\mathcal P^d$ is interesting on its own right (see Theorem \ref {th4.1} and Theorem \ref {th4.2}). Moreover, it helps to understand the \emph{localized} structure of the trajectory spaces $\mathcal T(v)$ for traversally generic fields $v$, the main subject of Theorem \ref {th5.2} and Theorem \ref {th5.3}.