It is known from for example Representations of Integers as Sums of Squares by Grosswald, E. that $$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ n_1^2+n_2^2+\cdots+n_k^2=N\}|\leq C_\varepsilon N^{\frac{k-2}{2}+\varepsilon}.$$ More generally, suppose we have a positive definite quadratic form of integer coefficients with possible linear terms $p(n_1,\ldots,n_k)$, say $p(n_1,n_2)=n_1^2+(n_2+1)^2+n_1(n_2+1)$, do we still have $$|\{(n_1,n_2,\ldots,n_k)\in\mathbb{Z}^k: \ p(n_1,\ldots,n_k)=N\}|\leq C_\varepsilon N^{\frac{k-2}{2}+\varepsilon}\ \ \ \ \ \ \ \ \ \ ?$$ I also know of another related fact (The number of integral points on arcs and ovals by E. BOMBIERI and J. PILA): if $\Gamma\subset\mathbb{R}^2$ is a real analytic image of the circle, then $|t\Gamma\cap \mathbb{Z}^2|\leq C_\varepsilon t^\varepsilon$. This could be used to prove special cases of the above inequality, say when $\Gamma=\{X^2+5Y^2=1\}$.

Edit: Also, is it possible to remove the $\varepsilon$ for $k\geq 5$? This is the case for counting representations of integers as sums of squares.