For Boolean polynomials in $\mathbb{Z}_p$ of sufficiently low degree we derive a relation expressing their values on one level set in terms of their values on another level set. We use this relation to derive linear upper and lower bounds, tight to within constant factor, on the degrees of various approximate majority functions, namely, functions that take the value 0 on one level set, the value 1 on a different level set, and arbitrary 0-1 values on other Boolean inputs. We show sublinear upper bounds in the case of moduli that are not prime powers.