I'm interested in this function

$$ h(m,n) = \operatorname{sgn}\Bigl(\lim_{w \to \infty}{\sum_{x_1=1}^{w}\dotsi\sum_{x_u=1}^{w}\dfrac{1}{1 + wp^2(m,n,x_1,\dotsc,x_u)}}\Bigr) $$

where $p$ is a polynomial in $u+2$ variables and integer coefficients. Can we simplify it to $$ h(m,n) = \operatorname{sgn}\Bigl(\dfrac{f(m,n)}{g(m,n)}\Bigr) $$

without the limit?