  Under mild conditions on $n,p$, we give a lower bound on the number of
$n$-variable balanced symmetric polynomials over finite fields $GF(p)$, where
$p$ is a prime number. The existence of nonlinear balanced symmetric
polynomials is an immediate corollary of this bound. Furthermore, we conjecture
that $X(2^t,2^{t+1}l-1)$ are the only nonlinear balanced elementary symmetric
polynomials over GF(2), where $X(d,n)=\sum_{i_1<i_2<...<i_d}x_{i_1} x_{i_2}...
x_{i_d}$, and we prove various results in support of this conjecture.
