I'm interested in this function $$ h(m,n) = \lim_{w \to \infty}\dfrac{1}{1+\prod_{x_1,...,x_u=1}^{w}p(m,n,x_1,...,x_u)} \\ $$ $p$ is a universal Diophantine polynomial with $2$ parameter and $u$ variables

before edit

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

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?