Polynomial Identity Testing Evaluating a polynomial on a circuit
If you're working over $\mathbb{Z}/p\mathbb{Z}$, you don't have to do anything: the intermediate results can't get larger than $p$.

If you're working over $\mathbb{Z}$, the standard trick is to pick a random $k$-bit prime $p$ and then reduce everything modulo $p$, so you are effectively working over $\mathbb{Z}/p\mathbb{Z}$.  One can show that the probability of getting a wrong answer using this method is exponentially small in $k$ (using the prime number theorem).

If you're working over $\mathbb{Q}$, you can reduce this to working over $\mathbb{Z}$ by clearing all fractions.