Plenty of Morse functions by perturbing with sums of squares

TL;DR

This paper proves that for a smooth function on R^n and a submanifold, the set of diagonal quadratic forms making the sum Morse when restricted to the submanifold is dense, using a refined approach beyond standard transversality.

Contribution

It establishes the density of diagonal quadratic forms that render the sum with a given function Morse on a submanifold, overcoming limitations of standard transversality methods.

Findings

The set of suitable quadratic forms is dense in the space of diagonal quadratic forms.

A refined approach is developed to handle cases where standard transversality fails.

The result applies to smooth functions and submanifolds in R^n.

Abstract

Given a smooth function f on R^n and a submanifold M, we prove that the set of diagonal quadratic forms q such that the restriction of f+q to M is Morse is a dense set (in the n-dimensional space of diagonal quadratic forms). The standard transversality argument seems not to work and we need a more refined approach.