Question
I would like to show that satisfying $\sum_{i=1}^{n}{x_i^{y_i}}=r$ is NP-Complete.
Consider $L= \{(\bar{y},r):\exists \bar{x} \text{ such that } \sum_{i=1}^{n}{x_i^{y_i}}=r\}$.
Where $\bar{x}=(x_1,x_2,\dots x_n)$ and $\bar{y}=(y_1,y_2,\dots y_n)$ are tuples of non-negative integers.
How long does it take to "decide" whether a given $(\bar{w},r)$ is in the "language" $L$? This is obviously in NP but is it NP-Complete remains to be concluded. Is this language NP-Complete?
Motivations
This problem comes out of my own explorations. I have been exploring solutions on hyperspheres and this is a natural generalization of a hypersphere. Note that $(<2,2,2,2,2,2>, 296675)\in L$ because by Lagrange we know that there exists some $(x_1,x_2,x_3,x_4,x_5,x_6)$ which satisfies $x_1^2+x_2^2+x_3^2+x_4^2+x_5^2+x_6^2=296675$. An $n>3$ dimensional hypersphere with radius $\sqrt{r}$, where $r$ is a whole number centered at the origin can always be satisfied by some integer coordinates (Lagrange's sum of 4 squares theorem). This makes the "decision" problem easy. There IS some solution though it might be hard to find.
But what about $x_1^2+x_2^3+x_3^4+x_4^5+x_5^6+x_6^7=296675$?
"Is there a solution to the equation above?" is the same as "Is $(<2,3,4,5,6,7>,296675) \in L$?" in my notation above.
This may be a computationally tricky problem to solve. I know of no guarantee that there must be some $(x_1,x_2,x_3,x_4,x_5,x_6)$ which satisfies the equation. On the otherhand, given an "instance" $(x_1,x_2,x_3,x_4,x_5,x_6)=(1,2,3,4,5,6)$ this problem is easy to verify. "Hard to solve but easy to verify" is a hallmark of NP problems.