Restrictions of nondegenerate Boolean functions and degree lower bounds over different rings

A Boolean function f : {0, 1}ⁿ → {0, 1} is called nondegenerate if f depends on all its n variables. We show that, for any nondegenerate function f, there exists a variable x_i such that at least one of the restrictions f_Ixi=0 or f_Ixi=1 must depend on all the remaining n – 1 variables. We also consider lower bounds on the degrees of polynomials representing a Boolean function over different rings. Let d_q(f) be the degree of the (unique) polynomial over the ring ℤ_q exactly representing f. For distinct primes p_i let m = Π^r_i=1 p^ei_i. Then, we show that any nondegenerate symmetric Boolean function f must have m · d_p1e₁(f)…d_pre_r(f) > n. We use the existence of nondegenerate subfunctions to prove degree lower bounds on random functions. Specifically, we show that m · d_p1e₁(f)…d_pre_r(f) > lg n – 1 holds for almost all f when f is chosen uniformly at random from all n-variate Boolean functions. Our proof uses the second moment method to show that a random f must almost always contain a nondegenerate symmetric subfunction on at least lg n – 1 variables. It follows that an n-variate nondegenerate symmetric Boolean function can have degree o(√(n)) over at most one finite field and that almost all f can have degree o(√(lg n)) over at most one finite field.