We prove lower bounds on the number of zeros of some classes of multivariate polynomials over $GF[q]$ in the function of the number of their terms only. The paper was motivated by some algebraic problems arising from the new randomized approximation techniques of [Karpinski,~Luby~91] and [Karpinski,~Lhotzky~91] for the number of zeros of polynomials over finite fields. %The bounds are expressed in the function of the %number of their terms only.