I'm interested in this function

$$ h(m,n) = \mathrm{sgn}\Big(\lim_{\substack{\varepsilon \to 0 \\ w \to \infty}}{\sum_{x_1=1}^{w}...\sum_{x_u=1}^{w}\dfrac{\varepsilon^2}{\varepsilon^2 + p^2(m,n,x_1,...,x_u)}}\Big) $$

where $p$ is a polynomial in $u+2$ variables. How can we simplify it?