Morse numbers of complex polynomials

TL;DR

This paper introduces algebraic formulas to compute the number of Morse points in a general linear Morsification of complex polynomials, linking singularity invariants to Morse trajectory counts.

Contribution

It provides new algebraic formulas for counting Morse points in complex polynomials with arbitrary singularities, connecting singularity invariants to Morse theory.

Findings

Algebraic formulas for Morse point counts derived

Explicit relations between singularities and Morse trajectories established

Applicable to polynomials with arbitrary singularities

Abstract

To a complex polynomial function $f$ with arbitrary singularities we associate the number of Morse points in a general linear Morsification $f_{t} := f - t\ell$. We produce computable algebraic formulas in terms of invariants of $f$ for the numbers of stratwise Morse trajectories which abut, as $t\to 0$, to some point of the space or at infinity.