Abstract: Let q be a power of a prime p, Fq be the finite field with q elements, and Fq[x1,...,,xn] be the ring of polynomials in n variables over Fq. The construction and study of local permutation polynomials of Fq[x1,...,,xn] is recently increasing interest among the experts. In this work, we study local permutation polynomials of maximum degree n(q-2) defined over the prime finite field Fp. IIn particular, we explicitly construct families of such polynomials when p[ mayor o igual que] 5 and n[ menor o igual que] p-1; and for any q of the form q=ppr when r [ mayor o igual que1] and p[ mayor o igual que] 3.
