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(2t,2t+1l−1) are the only nonlinear balanced elementary symmetric
polynomials over GF(2), where X(d,n)=∑i1<i2<...<idxi1xi2...xid, and we prove various results in support of this conjecture.Comment: 21 page