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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.

About how many connected components does this open subset have for large $d$ and/or $n$?

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If you consider the case $n=2$, I believe the precise answer to your question should be known for all $d$. V.I. Arnol'd was interested in such type of questions, originally in the case when you replace $S^1$ by $\mathbb R^1$ and homogeneous polynomials by inhomogeneous. Probably one should chase the references to Arnold's article : mathnet.ru/php/… You might also want to have a look on the article of Barannikov "On the space of real polynomials without multiple critical values" –  Dmitri Mar 30 '13 at 10:29
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1 Answer

up vote 25 down vote accepted

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. 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 who conjectured that
$$\log m(k)\sim 2k\log k $$

as $k\to \infty $. I refer you to this paper 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 and this paper.

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.

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Do you know any upper or lower bounds, even very bad ones? –  Will Sawin Apr 3 '13 at 18:28
    
The brief answer is I don't have an explicit upper bound on the top of my head but I think I know methods of producing a (really bad) upper bound valid for all polynomials $p\in U_d$ situated outside a semialgebraic subset of $U_d$ of codimension $\geq 1$. This is what many real algebraic geometers do for a living. I recommend a book by Basu, Pollack and Roy for inspiration perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted1.html –  Liviu Nicolaescu Apr 3 '13 at 20:49
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