LOCAL PERMUTATION POLYNOMIALS IN THREE VARIABLES OVER Zp

If p is a prime, let Zp denote the integers modulo p and Z* the set of nonzero elements of Zp. It is well known that every function from Zp x Zp x Zp into Zp can be represented as a polynomial of degree < p in each variable. We say that a polynomial f(x19 x29 x3) with coefficients in Zp is a local permutation polynomial in three variables over Zp if f(x19a9 b) 9 f{c9x29 d) , and f(e9f9 x$) are permutations in x±9 x2, and x3, respectively, for all a, b9 c9 d9 e 9 f e Zp. A general theory of local permutation polynomials in n variables will be discussed In a subsequent paper. In an earlier paper [2], we considered polynomials in two variables over Zp and found necessary and sufficient conditions on the coefficients of a polynomial in order that it represents a local permutation polynomial in two variables over Zp. The number of Latin squares of order p wds thus equal to the number of sets of coefficients satisfying the conditions given in [2]. In this paper, we consider polynomials in three variables over Zp and again determine necessary and sufficient conditions on the coefficients of a polynomial in order that it represents a local permutation polynomial in three variables over Zp. As in [1], a Latin cube of order n is defined as an n x n x n cube consisting of n rows, n columns, and n levels in which the numbers 0, 1, ..., n 1 are entered so that each number occurs exactly once in each row, column, and level. Clearly the number of Latin cubes of order p equals the number of local permutation polynomials in three variables over Zp. We say that a Latin cube is reduced if row one, column one, and level one are in the form 0, 1, ..., n 1. The number of reduced Latin cubes of order p will equal the number of sets of coefficients satisfying the set of conditions given in Section 2. In Section 3, we use our theory to show that there is only one reduced local permutation polynomial in three variables over Z3 and, thus, there is precisely one reduced Latin cube of order three.