So say I have a $n$ dimensional polynomial of degree $m$. Assume that $n\geq m$. Now for a degree $m$ polynomial in one dimension, we know that there can be at most $m-1$ local extrema. Is there a similar rule for multidimensional polynomials?
One important thing to note is that the polynomial I am working with is only linear terms of each dimension, so there will never be a $x_k^y$ for any $y>1$, for example
Good: $f(\vec x) = x_1x_2x_3 + x_1x_2 - x_3$
Bad: $f(\vec x) = x_1^2x_2 + x_3^3 - x_2$
If $f(x_1,\ldots,x_n)$ is a polynomial of degree $m$, then the critical points are the intersection of the $n$ sets $$V_i = \left\{\vec x \in \mathbb{R}^n \middle| \frac{\partial f}{\partial x_i}(\vec x) = 0\right\}$$ for $i = 1,\ldots,n$. Now $\partial f/\partial x_i$ has degree at most $m-1$, so Bézout's theorem tells you that the intersection has either infinitely many points or at most $$\prod_{i=1}^n \deg \frac{\partial f}{\partial x_i} \le (m-1)^n$$ points. If there are infinitely many points, then either $\partial f/\partial x_i \equiv 0$ for some $i$ or $\partial f_i/\partial x_i$ and $\partial f_j/\partial x_j$ have a common factor (i.e. $V_i$ and $V_j$ have a common component) for some $i\ne j$.
So $f$ has either infinitely many critical points or at most $(m-1)^n$ critical points.
