Let $f$ be a homogeneous polynomial of degree $d$ in $n$ variables. Restricted to the unit sphere $S^{n-1}$, it might or might not be a Morse function.


If $f$ is a Morse function of degree $1$, you get everyone's favorite Morse function on the sphere, with two critical points, corresponding to the minimal CW decomposition.


If $f$ is a Morse function of degree $2$, you get everyone's second-favorite Morse function, corresponding to the minimal regular CW decomposition with two points, two lines, two faces, and so on.


If $f$ has degree $d\geq 3$, there are a lot more possibilities. How many?


The space of degree $d$ homogeneous polynomials in $n$ variables can be identified with $\mathbb R^N$ where $N={\left( \begin{array}{c}n+d-1 \\ n-1\end{array}\right)}$. The ones that are Morse functions form an open subset.



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> About how many connected components does this open subset have for large $d$ and/or $n$?
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How many polynomial Morse functions on the sphere?
I've been trying to answer this question for several years and it turned out to be really hard, even for the $2$-sphere. Below I will discuss this case.


First of all one should ask what is the number $m(k)$ of topological types of (stable) Morse functions on $S^2$ with precisely $k$ saddle points. (such a function has $2k+2$ critical points.) I showed that the generating series


$$ x(t) := \sum\_{k\geq 0} \frac{m(k)}{(2k+1)!} t^{2k+1}, $$


is the inverse of an elliptic integral; see [this paper](http://www3.nd.edu/%7Elnicolae/Morse-count.pdf). More precisely $x(t)$ is the inverse of the function


$$ x\mapsto t(x)=\int\_0^x \frac{ds}{\sqrt{s^4/4-s^2-2sx+1}} ds. $$


This fact leads to a positive answer to a [question of V.I. Arnold](http://www.mathnet.ru/php/presentation.phtml?option_lang=eng&presentid=189) who conjectured that  

$$\log m(k)\sim 2k\log k $$


as $k\to \infty $. I refer you to [this paper](http://www3.nd.edu/%7Elnicolae/statistics.pdf) for details. This shows that $m(k)$ grows rather fast as $k\to \infty$.


Any polynomial $P$ of degree $d$ in $\newcommand{\bR}{\mathbb{R}}$ on $\bR^n$ can be uniquely decomposed as a sum


$$ P= \sum\_{0\leq j+2k\leq d} r^{2k} H\_{j}, \;\; r^2= (x\_1^2+\cdots +x\_n^2), $$


where $H\_{j}$ is a darmonic polynomial of degree $j$. On $\bR^3$ the space of degree $d$ hormonic polynomials has dimension $2d+1$. If we denote by $U\_d$ the subspace of $C^\infty(S^2)$ consisting of the restrictions to $S^2$ of the polynomials of degree $\leq d$ we deduce that


$$\dim U\_d=\sum\_{0\leq k\leq d} (2k+1)=(d+1)^2. $$


Denote by $K\_d$ the expected number of critical points of a random function in $U\_d$. I showed that


$$ K\_d\sim C\dim U\_d\sim Cd^2 $$


as $d\to \infty$ where $C$ is a certain explicit constant; see [this paper](https://arxiv.org/abs/1008.5085) and [this paper](http://www3.nd.edu/%7Elnicolae/CritSetStat.pdf).


It turns out that the number of critical points of a random function in $U\_d$ is highly concentrated around its mean $K\_d$, i.e., the probability that the number of critical points of a random function in $U\_d$ is far from the mean $K\_d$ is extremely small as $d\to\infty$. In more precise technical terms, the variance of the (random) number of critical points of a (random) function in $U\_d$ has the same size as $K\_d$, which makes the standard deviation of size $\sqrt{K\_d}$, much, much smaller than $K\_d$ for $d$ large.


I personally believe, based on some empirical evidence, that the mean is close to the maximum number of critical points in the sense that if we denote by $\mu\_d$ the maximum number of critical points of a Morse function in $U\_d$, then $\mu\_d \sim C'' d^2$ as $d\to\infty$.


My guess is that the number of topological types of functions in $U\_d$ as $d\to \infty$ is roughly


$$ \sum\_{k=1}^{K\_d/2} m(k), $$


where I recall that $m(k)$ denotes the number of topological types of Morse functions with $k$ saddle points, i.e., $2k+2$ critical points.


