Let $f_1, \dots, f_n$ be a finite set of polynomials in the polynomial ring $Z[x_1, \dots, x_m]$. At a prime $p$, let $N_p$ be the number of solutions $x=(x_1, \dots, x_m)\in (\mathbb{Z}/p\mathbb{Z})^m$ to the system $$f_1(x)=f_2(x)=f_3(x)=\dots=f_n(x)=0$$ (with coordinates $x_1, \dots, x_m\in \mathbb{Z}/p\mathbb{Z}$).

As a function of $p$, are there any conditions (on the set of polynomials $f_1, \dots, f_n$) under which $N_p$ is a polynomial in $p$ for large enough values (or perhaps all values) of $p$?

The system that I'd like this to be true for (all primes $p$) is $$f_1=(a_3^2-a_3)-a_2(c_1^2-c_1)-a_1(b_1^2-b_1)=0$$ $$f_2=(a_4^2-a_4)-a_2(c_2^2-c_2)-a_1(b_2^2-b_2)=0$$ $$f_3=a_3a_4-a_2c_1c_2-a_1b_1b_2=0 $$

in the $8$ variable polynomial ring $\mathbb{Z}[a_1,a_2, a_3,a_4, b_1, b_2, c_1, c_2]$.

Computations done for $p\leq 19$ lead us to expect $N_p=p^5 + 12 p^4 - 20 p^3 + 30 p^2 - 10 p$.