I try to solve the finite system of multivariable algebraic equations with coefficients from $\mathbb Q$. It would be sufficient for me to prove that there is only finite number of solutions over $\mathbb Q$ (or even more over algebraic closure $\bar{\mathbb Q}$).

I try to do it by calculating dimension of corresponding ideal (my goal is to get zero dimension). But direct calculation is too demanding due too growths of the rational height of coefficients.

Are there any local-global principle in that case?

My idea is to calculate dimension of ideal based on reduced system over $\mathbb F_p$ for several primes $p$. Then get for example zero dimension ideal and then conclude that dimension of corresponding ideal over $\bar{\mathbb Q}$ is therefore zero. Are there theorems and approaches helping with that?

P.S. Example of system (variables: a0, a1, a2):

```
5/2*a0*a2^4 - 6*a1*a2^4 + 27/4*a1^2*a2^2 + 9/4*a2^4 = 0
243/32*a0*a1*a2^2 + 405/32*a0^2*a1 - 243/64*a0^2 = 0
27/8*a1^4*a2^4 - 135/8* a0*a1*a2^6 + 9*a1^2*a2^6 + 9/4*a2^8 - 9/4*a0^2*a1^5 = 0
```