Morse numbers and the Morse polynomial

Let $M$ be a closed smooth $n$-manifold (Smooth manifolds and their smooth charts, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right) and let $f:M\to \mathbb{R}$ be a Morse function (Morse functions and excellent Morse functions). Write $\mathrm{Crit}(f)$ for the set of critical points of $f$ (Critical points and critical values of a smooth function) and $\mathrm{ind}(p)$ for the index of a nondegenerate critical point $p$ (Nondegenerate critical points, nullity, index, and coindex). For every integer $k$ the Morse number of $f$ in degree $k$ is $m_k(f):=\mathrm{\#}\{p\in \mathrm{Crit}(f):\mathrm{ind}(p)=k\}\mathrm{.}$

Each $m_k(f)$ is a finite nonnegative integer, and $m_k(f)=0$ unless $0\le k\le n$. The Morse polynomial of $f$ is $M_f(t):={\sum }_{k=0}^n{m}_k(f)t^k\in \mathbb{Z}[t],$ a polynomial with nonnegative integer coefficients and $M_f(1)=\mathrm{\#}\mathrm{Crit}(f)$.

The index is the number of negative squares of the Hessian in the library's convention (Nondegenerate critical points, nullity, index, and coindex). The definition is purely geometric: no field, coefficient ring, or orientation enters, and the empty and zero-dimensional cases are included.

• Well-definedness. Finiteness of each $m_k(f)$ is A Morse function on a compact manifold has finitely many critical points: a Morse function on a compact manifold has only finitely many critical points, so the displayed cardinality is a nonnegative integer. The index of a nondegenerate critical point is an integer in $[0,n]$ because it is the number of negative squares of a symmetric bilinear form on the $n$-dimensional space $T_{p}M$ (Nondegenerate critical points, nullity, index, and coindex); this is why the sum defining $M_f$ is finite and why $M_f$ has degree at most $n$.
• Conventions. A Morse function on a closed manifold is a Morse function in the sense of Morse functions and excellent Morse functions on a compact manifold without boundary; no excellent condition is required here. The zero-dimensional case $n=0$ is the case of a finite set of points, where $m_0(f)=\mathrm{\#}M$ and $M_f(t)=\mathrm{\#}M$.
• The Morse numbers depend only on $f$, not on any field; the $F$-Betti numbers compared with them below do depend on the coefficient field (Poincare polynomial of a space and of a pair over a field).