For $p \in \mathbb{R}$, consider the function $$F_p(\lambda_1, \dots, \lambda_n) = \lambda_1^p + \dots + \lambda_n^p.$$ My goal is to maximize this function under the constraints that $$ \lambda_1^2 + \dots + \lambda_n^2 = 1, ~~~\text{and}~~~ \lambda_1 + \dots + \lambda_n = 0.$$ I am mainly interested in the case $p \in \mathbb{N}$, $p \geq 3$. In that case $F_p$ is a polynomial.

**Remarks**

- If $p=3$, one can show that the maximum is achieved at the vector $(a, -b, \dots, -b)$, where $a, b>0$ are fixed by the constraints. A working hypothesis is that this is true for any $p$. I do not know how to treat $p = 4, 5, \dots$ though.
- The reason is that for $p=3$, one can use the method of Lagrange multipliers to get that each $\lambda_j$ satisfies the same quadratic equation, hence the maximizer consists of vectors $(\lambda_1, \dots, \lambda_n)$, where $\lambda_j = a$ for $k$ indices $j$ and $\lambda_j = b$ for $n-k$ indices $j$. The optimal $k$ can then be easily determined. For general $p \in \mathbb{N}$, one gets that the $\lambda_j$ satisfy a polynomial equation of degree $p-1$, hence take one of $p-1$ values. Already for $p=4$, this makes matters quite hard.
- Of course, maximizing a polynomial over a sphere is very hard in general, but this is a very basic, very explicit polynomial, so one could hope to have an explicit exact solution.
- The problem is equivalent to maximizing $$\tilde{F}_p(A) = \mathrm{tr}(A^p)$$ over the space of symmetric trace-free matrices of norm one, hence to maximize an $O(n)$-invariant polynomial over an irreducible $O(n)$-representation.

**Related questions**

- If $p$ is odd, minimizing and maximizing is the same thing, but if $p$ is even, one could also ask to minimize the value of $F_p$.
- What if one drops the second constraint? [\edit: This is uninteresting: The maximum is one, taken at a unit vector.]