I have a multivariate polynomial function of N variables
$f(x_1,x_2,…,x_N) = x_1 x_2 x_3 .. x_N \left( 1 + \sum_i^N (a_i x_i^2 - x_i) \right)$,
where $a_i > 0$ are real positive numbers.
By Bezout's theorem, this has $2^N$ critical points, which may be complex.
I'm interested in the critical points for real positive $x_i$, so there are at most $2^N$ of them, but there may be fewer.
Here's my question:
Can I make a stronger statement? That there are at most one minimum, at most N one-negative mode saddle points, at most $N$-choose-$2$ two-negative mode saddle points, etc.?