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.