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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.?

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  • $\begingroup$ What does 'categorizing' mean in your title? and what's the relation of your question to algebraic topology? $\endgroup$ Nov 17, 2013 at 10:33
  • $\begingroup$ By "categorizing" I meant categorizing the critical points by their number of negative modes, as in the final sentence of my question. I want to know whether it is possible to bound not just the total number of saddle points, but the number of saddle points with a given number of negative modes. $\endgroup$
    – Adam B
    Nov 17, 2013 at 20:26
  • $\begingroup$ I've changed the tag to "critical-point-theory". $\endgroup$
    – Adam B
    Nov 17, 2013 at 20:26

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