On Some Permutation Trinomials in Characteristic Three

Year 2024, Early Access, 1 - 10

Burcu Gülmez Temür , Buket Özkaya

https://doi.org/10.15672/hujms.1443686

Abstract

In this paper, we determine the permutation properties of the polynomial x^3+x^q+2−x^4q−1
over the finite field Fq2 in characteristic three. Moreover, we consider the trinomials of
the form x^4q−1 +x^2q+1 ±x^3. In particular, we first show that x^3 +x^q+2 −x^4q−1 permutes
F_q^2 with q = 3^m if and only if m is odd. This enables us to show that the sufficient
condition in [35, Theorem 4] is also necessary. Next, we prove that x^4q−1 + x^2q+1 − x^3
permutes F_q^2 with q = 3^m if and only if m is congruent to 0 modulo 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 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.

Keywords

Permutation polynomials, Finite fields, Absolutely irreducible

References

• [1] A. Akbary and Q. Wang, On polynomials of the form x rf(x (q−1)/l), Int. J. Math. Math. Sci., Art. ID 23408, 2007.