Relate solutions to a polynomial system in complex numbers to solutions in a finite field

Question

Suppose I have a system of polynomials which are homogeneous but of distinct degrees that I want to solve simultaneously:

$$F_1(z_1,\ldots,z_n)=\cdots=F_m(z_1,\ldots,z_n)=0.$$

Let $X(\mathbb F)$ denote the solutions to this system over the field $\mathbb F$. For the application I have in mind, the system is overdetermined so $m>n$. Now suppose I can show something like $X(\mathbb C)$ contains no non-trivial points, or that the points all live in some other variety, or some other strong type of statement about the complex solutions. Is there a condition under which a similar result could be deduced for $X(\mathbb F_p)$? Here $\mathbb F_p$ is the prime field with $p$ elements.

Answer 1

If $X(\mathbb{C})$ is empty, or all solutions satisfy some further equation $G(x_1, \ldots, x_n)=0$, then the same holds true for $X(\mathbb{F}_p)$, provided that $p$ is sufficiently large. To see this note that you can check these properties algorithmically by computing resultants, Gröbner bases, or Macaulay determinants. In any such computation you will encounter only a finite number of integers which you have to invert, thus, when passing to $\mathbb{F}_p$ the computation is still valid, unless $p$ divides such an integer. Thus there is a finite, computable set of primes, such that all properties remain over $\overline{\mathbb{F}_p}$, and a forteriori over $\mathbb{F}_p$.

In reality computing these exceptional primes can be extremely time consuming. As the number of variables grows, the problem can become untractable quite fast, even if the polynomials are only quadratic.