Abstract. Let $p$ be a prime, and $F(x_1,x_2,\ldots, x_n)\in{\msbm F} _p[x_1,\ldots,x_n]$ be a nonconstant polynomial such that the degree of $F$ in each variable $x_i$ is at most $p-1$. The {\it rank} of $F$ is the least integer $r$ for which there exists an invertible homogeneous linear change of variables which carries $F$ into a polynomial with precisely $r$ variables. In [6] Rédei proposed the following conjecture: if the rank of $F$ is at least $\deg F$, then the equation (congruence) $F(x_1,\ldots,x_n)=0$ has a solution in ${\msbm F} _p^n$. We disprove the conjecture by giving counterexamples. On the other hand, we show that it holds for some important special cases, including generalized diagonal equations.

AMS Subject Classification (1991): 11T06, 11D79

Keyword(s): Finite fields, equations, solvability

Received July 29, 2002, and in revised form April 30, 2003. (Registered under 2914/2009.)