Local Permutation Polynomials of Maximum Degree Over Prime Finite Fields

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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. In particular, we explicitly construct families of such polynomials when p≥5 and n≤p-1; and for any q of the form q=ppr when r≥1 and p≥3.