This question follows the article discussed here

## Problem

Suppose we're trying to bound the number of integral solutions to a system of multi-variable polynomials, say

$$ \sum_{i=1}^n x_i^t = \sum_{i=1}^n y_i^t, $$ where each $x_i,y_i \in \mathbb N$ and for each $t < c$ for some constant $c$.

If we do not put any constrains on the solution, there are infinitely many possible solutions even when $n=C=1$. So if we put some constrains on {$x_i,y_i$} like $x_i,y_i \in$ {$0,1,\ldots,n$}, then how many possible solutions can we get? Naively there are $O(n^n)$ choices, but it seems highly unlikely that there are many solutions to the system of equations. Is there any exist bound on the number of solutions, say $O(n^k)$ for fixed $k$ or even better bounds? Are there some well-known approaches to bound the number of solutions of an equation?

## Motivation

This question arose when I'm trying to come up with some reasonable constrains with the equation in Prouhet-Tarry-Escott Problem. It seems like if we restrict the maximum value of variables, there aren't many solutions to the equation. I tried to add more constrains to get rid of the already few solutions, but it seems that there is no direct way making the solution set empty, that is, no possible solutions under such constrains.

So I turn to find some existing bounds for the equation, but sadly nothing occurred. Can it be still hard to find such results, or there are some theorems like the Fundamental Theorem of Algebra, concerning the number of solutions to a multi-variable equation? Any information is useful. Thank you all!

## Edited

According to Felipe Voloch (Thanks!), the general approach to the question is the Hardy-Littlewood method, which considers the number of solutions to an equal-power Diophantine equation. But it seems that the method gives a lower bound on the number of solutions (is this correct?), rather than an upper bound. Or there are some ways to give upper bounds by the same method?

One more question: How about further restricting the solutions to be prime numbers? Does this make any difference?