Hacettepe Journal of Mathematics and Statistics 2651-477X 2651-477X Hacettepe University 10.15672/hujms.1443686 Algebra and Number Theory Cebir ve Sayı Teorisi On some permutation trinomials in characteristic three https://orcid.org/0000-0002-0435-6894 Gülmez Temür Burcu ATILIM UNIVERSITY https://orcid.org/0000-0003-2658-5441 Özkaya Buket MIDDLE EAST TECHNICAL UNIVERSITY, INSTITUTE OF APPLIED MATHEMATICS 06 24 2025 54 3 797 806 02 27 2024 06 11 2024 Copyright © 2002, Hacettepe Journal of Mathematics and Statistics 2002 Hacettepe Journal of Mathematics and Statistics

In this paper, we determine the permutation properties of the polynomial $x^3+x^{q+2}-x^{4q-1}$ over the finite field $\mathbb{F}_{q^2}$ in characteristic three. Moreover, we consider the trinomials of the form $x^{4q-1}+x^{2q+1} \pm x^{3}$. In particular, we first show that $x^3+x^{q+2}-x^{4q-1}$ permutes $\mathbb{F}_{q^2}$ with $q=3^m$ 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 $x^{4q-1}+x^{2q+1} - x^{3}$ permutes $\mathbb{F}_{q^2}$ with $q=3^m$ if and only if $m\not\equiv 0 \pmod 4$. Consequently, we prove that the sufficient condition in [20, Theorem 3.2] is also necessary. Finally, we investigate the trinomial $x^{4q-1}+x^{2q+1} + x^{3}$ and show that it is never a permutation polynomial of $\mathbb{F}_{q^2}$ in any characteristic. All the polynomials considered in this work are not quasi-multiplicative equivalent to any known class of permutation trinomials.

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