Balanced Symmetric Functions over $GF(p)$

Abstract

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.

21 pages