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 and integer coefficients. can we simplify it to
$$
h(m,n) = sgn(\dfrac{f(m,n)}{g(m,n)})
$$

without the limit