Minimizing...

A friend asked me this math question (presumably popped out of QFT or something)

Consider a set of N elements: {a_i}
It is known that sum(a_i) = 0
and that sum(a_i²) = 1
What is the minimum of sum(a_i⁴)?

The AM-RMS/power mean inequality tells us that sqrt(sum(a_i⁴)/N) >= sum(a_i²)/N ->
sum(a_i⁴) >= 1/N, with equality iff the a_is are the same.

If N is even, then the a_is just alternate as +-1/sqrt(N)
If N is odd, then that trick doesn't work, since then the sum won't = 0.  Therein lies the problem.  It seems likely that the answer has (N+1)/2 of the a_is equaling one number, and the remainder equaling something else.  However, I haven't been able to prove that this is the optimal solution yet.
I tried brute forcing this for N=3, but after an hr of algebra... the answer is always 1/2, regardless of how I pick the a_is.

>>4
The rest of them check out as well, so it looks like that's it.

If you want complex solutions, you have to specify wha exactly you're minimizing, since the sum will be complex in general.