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On Some Permutation Trinomials in Characteristic Three
(Hacettepe Univ, Fac Sci, 2025) Temür, Burcu Gülmez; Özkaya, Buket; Mathematics; 02. School of Arts and Sciences; 01. Atılım University
In this paper, we determine the permutation properties of the polynomial x3 +xq+2 −x4q−1 over the finite field Fq2 in characteristic three. Moreover, we consider the trinomials of the form x4q−1 + x2q+1 ± x3. In particular, we first show that x3 + xq+2 − x4q−1 permutes Fq2 with q = 3m if and only if m is odd. This enables us to show that the suﬃcient condition in [34, Theorem 4] is also necessary. Next, we prove that x4q−1 + x2q+1 − x3 permutes Fq2 with q = 3m if and only if m ̸≡ 0 (mod 4). Consequently, we prove that the suﬃcient condition in [20, Theorem 3.2] is also necessary. Finally, we investigate the trinomial x4q−1 + x2q+1 + x3 and show that it is never a permutation polynomial of Fq2 in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.