Temür, Burcu GülmezÖzkaya, Buket2025-08-052025-08-0520251303-50102651-477X10.15672/hujms.14436862-s2.0-105009256921https://doi.org/10.15672/hujms.1443686https://search.trdizin.gov.tr/en/yayin/detay/1321829/on-some-permutation-trinomials-in-characteristic-threehttps://hdl.handle.net/20.500.14411/10718In 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.eninfo:eu-repo/semantics/openAccessPermutation PolynomialsFinite FieldsAbsolutely IrreducibleArticle