Balanced Symmetric Functions Over ${\hbox {GF}}(p)$

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 prove that X(2^t, 2^t+1lscr-1) are balanced and conjecture that these are the only balanced symmetric polynomials over GF(2), where X(d, n) = Sigma_{1lesi 1 <i 2 <hellip<i d lesn}x_{i 1}x_{i 2}hellipx_{i d}.