Let $p(x_{1},x_{2},\ldots,x_{n})=\prod_{i,j=1}^{n}{a_{ij}x_{i}x_{j}}$ be a homogenous multivariate polynomial of degree $2$. 

> I would like to know **how many extrema** $p$ has on the standard simplex
> in $\mathbb{R}^{n}$ (i.e. the set $\{(x_{1},x_{2},\ldots,x_{n}) |
> \sum_{i=1}^{n}{x_{i}}=1\}$).

Are there standard ways to count or at least estimate the number of extrema? 

Without the simplex constraint it's (almost but not quite) the same as asking how many zeros the gradient has but apparently the constraint further complicates things.

Or am I missing something altogether obvious?

P.S.
[Bezout's theorem][1] comes to my mind but I can't quite make it apply to this case.


  [1]: https://en.wikipedia.org/wiki/B%C3%A9zout%27s_theorem