What's the Maximum Number of Local Minima an Nth-Order Polynomial Surface or Solid Can Have?

Question

I have an optimization problem that in the ideal case can be approximated by a 3D polynomial surface, i.e. $$f_{n}\left(x,y\right)=\sum _{i,j\leq{n}} C_{i,j}x^{i}y^{j}$$ My assumptions are:

1. All minima are encompassed over my scanned space along $x$ and $y$.
2. The data and apparent minima can be approximated well with a smooth surface, which is not flat.
3. For any $n^{th}$ order fit, the surface must have one or more global minima.

From criteria 2.), for instance, for $2^{nd}$ order (quadratic fits), we can disqualify all functions of the forms $C_{2,0}x^2-C_{0,2}y^2...$ and $C_{2,0}y^2-C_{0,2}x^2...$ (hyperbolic parabaloids). Likewise we can eliminate fits that lack either x or y terms and fits which produce imaginary solutions.

In the simple case, the surface can be approximated by a quadratic and probably looks something like this:

However, for more complicated systems, I expect my objective function to produce multiple relative minima. Further, in some cases I may need to expand my objective function into z as well, and fit to $f_{n}\left(x,y,z\right)=\sum _{i,j,k\leq{n}} C_{i,j}x^{i}y^{j}z^{k}$ with similar criteria (at edges goes to $\infty$ or constant value, never $-\infty$; has at least 1 global minima)

With that in mind, my questions are:

1. What is the maximum number of relative minima for $n^{th}$ polynomial surfaces meeting my criteria?
2. If fitting to $f_{n}\left(x,y,z\right)=\sum _{i,j,k\leq{n}} C_{i,j}x^{i}y^{j}z^{k}$ what is the maximum number of relative minima for $n^{th}$ polynomial solids meeting the criteria?

It seems like surely there is an elementary answer to the above. However, I've been having trouble finding pertinent discussions/lectures, save for a couple regarding $n^{th}$ order polynomials of a single variable, which can have a maximum of $n-1$ minima.

Also from some freeform exploration of polynomials on Wolfram Alpha, I know that you can at least have 2 minima for a $4^{th}$ order polynomial surface, such as $z=x^{4}+y^{4}-y^{3}+x^{3}-x^{2}-y^{2}-x-y$. (Update: $z=x^{4}+y^{4}-y^{3}+x^{3}-x^{2}-y^{2}$ has 4 minima, so clearly the maximum number of minima for an $n^{th}$ order polynomial must be at least $n$.)

Answer 1

The maximum number of local minima is infinite, at least for polynomial of degree more than $4$. For instance, consider the $4$ degree polynomial

$$ p(x,y) = (x^2 + y^2 - 1)^2.$$ Every point in the unit circle is a minimum of $p$ and there is an infinite amount of them (even non countable).