Let $f$ be a Boolean function.

Let $g$ be the minimum degree real polynomial that represents $f$ with degree $d$.

Let $g_{p}$ be the minimum degree $\Bbb F_p$ polynomial that represents $f$ with degree $d_p$.

Is $d_p\leq d$?

If $gcd(p_1,p_2)=1$, is there a relation between $d_{p_i}$s?

What is a good reference to understand relations among these degrees?

Would it be reasonable to query about approximate degrees over $\Bbb F_p$?