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: $f(\vec x) = x_1^2x_2 + x_3^3 - x_2$