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Let $p\in \left[0,1\right]$ and take $a_1,a_2,\ldots,a_n\in \mathbb{R}^{+}$. What is the maximum number of solutions that the system of (nonlinear) equations $x_1^p +x_2^p +\cdots+x_n^p = 1\\ x_1^{1-p}\left(a_1-x_1\right)=x_2^{1-p}\left(a_2-x_2\right)=\cdots =x_n^{1-p}\left(a_n-x_n\right),$ can have in $\left[0,a_1\right]\times \left[0,a_2\right]\times \cdots \times \left[0,a_n\right]$? Can we solve this system of equations in polynomial time?

I know that there are results in Algebraic Geometry that can give upper bounds on the number of solutions, at least for rational values of $p$, but they are for a generic system of polynomial equations; for the specific equations above those bounds seem to be very loose.

EDIT:

  • There is actually another condition that I forgot: for any $1\leq i,j\leq n$, if $a_i>a_j$ the solution must satisfy $x_i>x_j$. In other words, the $x_i$'s and $a_i$'s should have the same order.
  • I derived these equations while trying to solve a non-convex optimization problem.
  • Based on some numerical experiments, my conjecture is that the are no more than 3 solutions.
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    My mistake: I had remembered posynomial as allowing real exponents but no restrictions on the coefficients, which is what you're asking about. Regardless of the name, there are bounds for this type of equation, for example here: [link](http://arxiv.org/abs/math/0609544). This still gives an exponential bound, but maybe you could do better after rearranging the equation.2012-09-20

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